@article{petropavlovsky_tsynkov_turkel_2024, title={Computation of unsteady electromagnetic scattering about 3D complex bodies in free space with high-order difference potentials}, volume={498}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2023.112705}, abstractNote={We extend the previously developed high-order accurate method for acoustic scattering to electromagnetic scattering, i.e., from the scalar setting to a vector setting. First, the governing Maxwell's equations are reduced from their original first-order form to a system of second-order wave equations for the individual Cartesian components of electromagnetic field. In free space, these wave equations are uncoupled. Yet at the boundary of the scatterer, the variables that they govern (i.e., Cartesian field components) remain fully coupled via the boundary conditions that account for the specific scattering mechanism. Next, the wave equations are equivalently replaced with Calderon's boundary equations with projections obtained using the method of difference potentials and a compact high-order accurate scheme. The Calderon's boundary equations are combined with the boundary conditions and the overall system is solved by least squares. The resulting vector methodology (electromagnetic) inherits many useful properties of the scalar one (acoustic). In particular, it offers sub-linear computational complexity, does not require any special treatment of the artificial outer boundary, and has the capacity to solve multiple similar problems at a low individual cost per problem. We demonstrate the performance of the new method by computing the scattering of a given impinging wave about a double-cone hypersonic shape.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Petropavlovsky, Sergey and Tsynkov, Semyon and Turkel, Eli}, year={2024}, month={Feb} }
 @article{versano_turkel_tsynkov_2024, title={Fourth-Order Accurate Compact Scheme for First-Order Maxwell's Equations}, volume={100}, ISSN={["1573-7691"]}, DOI={10.1007/s10915-024-02583-5}, abstractNote={Abstract We construct a compact fourth-order scheme, in space and time, for the time-dependent Maxwell’s equations given as a first-order system on a staggered (Yee) grid. At each time step, we update the fields by solving positive definite second-order elliptic equations. We develop compatible boundary conditions for these elliptic equations while maintaining a compact stencil. The proposed scheme is compared computationally with a non-compact scheme and with a convolutional dispersion relation preserving (DRP) scheme.}, number={2}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Versano, I. and Turkel, E. and Tsynkov, S.}, year={2024}, month={Aug} }
 @article{gilman_tsynkov_2024, title={Modeling the Earth's Ionosphere by a Phase Screen for the Analysis of Transionospheric SAR Imaging}, volume={62}, ISSN={["1558-0644"]}, DOI={10.1109/TGRS.2023.3335146}, abstractNote={In the problems of transionospheric synthetic aperture radar (SAR) imaging, autofocus, and ionospheric tomography, the Earth’s ionosphere is often represented by a phase screen. A key advantage of the phase screen is that it reduces the overall dimension of the model. Yet, this convenient simplification comes at a price of introducing inaccuracies into the modeled quantities, such as the phase of the propagating radar signals. In this work, we develop the appropriate metrics to quantify these inaccuracies and evaluate their role for two particular scenarios: SAR imaging through large-scale ionospheric disturbances due to the atmospheric gravity waves (AGWs) and SAR imaging through ionospheric turbulence.}, journal={IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING}, author={Gilman, Mikhail and Tsynkov, Semyon}, year={2024} }
 @article{gilman_tsynkov_2023, title={Transionospheric Autofocus for Synthetic Aperture Radar}, volume={16}, ISSN={["1936-4954"]}, DOI={10.1137/22M153570X}, abstractNote={Turbulent fluctuations of the electron number density in the Earth’s ionosphere may hamper the performance of spaceborne synthetic aperture radar (SAR). Previously, we have quantified the extent of the possible degradation of transionospheric SAR images as it depends on the state of the ionosphere and parameters of the SAR instrument. Yet no attempt has been made to mitigate the adverse effect of the ionospheric turbulence. In the current work, we propose a new optimization-based autofocus algorithm that helps correct the turbulence-induced distortions of spaceborne SAR images. Unlike the traditional autofocus procedures available in the literature, the new algorithm allows for the dependence of the phase perturbations of SAR signals not only on slow time but also on the target coordinates. This dependence is central for the analysis of image distortions due to turbulence, but in the case of traditional autofocus where the distortions are due to uncertainties in the antenna position, it is not present.}, number={4}, journal={SIAM JOURNAL ON IMAGING SCIENCES}, author={Gilman, Mikhail and Tsynkov, Semyon V.}, year={2023}, pages={2144–2174} }
 @article{gilman_tsynkov_2023, title={Transionospheric Autofocus for Synthetic Aperture Radar}, ISSN={["2835-1355"]}, DOI={10.1109/ICEAA57318.2023.10297676}, abstractNote={Synthetic aperture radar (SAR) illuminates the target with microwaves and uses digital signal processing to build the image. For a spaceborne SAR, the antenna is mounted on a satellite, whereas the target area is on the ground. The antenna emits pulses of electromagnetic radiation and receives the returns, i.e., signals reflected off the target. The signal processing algorithm takes into account multiple pulses emitted and received by the antenna at a series of its successive positions, called the synthetic aperture. The resulting image approximates the backscattering reflectivity of the target. Mathematically, SAR imaging is equivalent to solving an inverse problem - that of reconstructing the unknown target reflectivity given radar returns as the input data.}, journal={2023 INTERNATIONAL CONFERENCE ON ELECTROMAGNETICS IN ADVANCED APPLICATIONS, ICEAA}, author={Gilman, Mikhail and Tsynkov, Semyon}, year={2023}, pages={24–24} }
 @article{gilman_tsynkov_2023, title={Vertical autofocus for the phase screen in a turbulent ionosphere}, volume={39}, ISSN={["1361-6420"]}, DOI={10.1088/1361-6420/acb8d6}, abstractNote={Abstract}, number={4}, journal={INVERSE PROBLEMS}, author={Gilman, Mikhail and Tsynkov, Semyon}, year={2023}, month={Apr} }
 @article{petropavlovsky_tsynkov_turkel_2022, title={3D time-dependent scattering about complex shapes using high order difference potentials}, volume={471}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2022.111632}, abstractNote={We compute the scattering of unsteady acoustic waves about complex three-dimensional bodies with high order accuracy. The geometry of a scattering body is defined with the help of CAD. Its surface is represented as a collection of non-overlapping patches, each parameterized independently by means of high order splines (NURBS). As a specific example, we consider a submarine-like scatterer constructed using five different patches. The acoustic wave equation on the region exterior to the scatterer is solved by first reducing it to a system of Calderon's boundary operator equations. The latter are obtained using the method of difference potentials coupled with a compact fourth order accurate finite difference scheme. When solving the boundary operator equations, we employ Huygens' principle. It allows us to work on a sliding time window of non-increasing duration rather than keep the entire temporal history of the solution at the boundary. The proposed methodology demonstrates grid-independent computational complexity at the boundary and sub-linear complexity with respect to the grid dimension. It efficiently handles complex non-conforming geometries on Cartesian grids with no penalty for either accuracy or stability due to the cut cells. Its performance does not deteriorate over arbitrarily long simulation times. The exact treatment of artificial outer boundaries is inherently built in. Finally, multiple similar problems can be solved efficiently at a low individual cost per problem. This is important when, for example, the boundary condition on the surface changes but the scattering body stays the same.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Petropavlovsky, Sergey and Tsynkov, Semyon and Turkel, Eli}, year={2022}, month={Dec} }
 @article{kahana_smith_turkel_tsynkov_2022, title={A high order compact time/space finite difference scheme for the 2D and 3D wave equation with a damping layer}, volume={460}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2022.111161}, abstractNote={We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. For unbounded domains we add a fourth order accurate sponge layer to damp the outgoing waves. We demonstrate that usually this is more efficient than lower order schemes despite being implicit and conditionally stable. Fast time marching of the implicit scheme is accomplished by iterative methods such as multi-grid. Computations confirm the design convergence rate for the in-homogeneous, variable wave speed equation.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Kahana, Adar and Smith, Fouche and Turkel, Eli and Tsynkov, Semyon}, year={2022}, month={Jul} }
 @article{north_tsynkov_turkel_2022, title={High-order accurate numerical simulation of monochromatic waves in photonic crystal ring resonators with the help of a non-iterative domain decomposition}, volume={11}, ISSN={["1572-8137"]}, DOI={10.1007/s10825-022-01973-y}, journal={JOURNAL OF COMPUTATIONAL ELECTRONICS}, author={North, Evan and Tsynkov, Semyon and Turkel, Eli}, year={2022}, month={Nov} }
 @article{north_tsynkov_turkel_2022, title={Non-iterative domain decomposition for the Helmholtz equation with strong material discontinuities}, volume={173}, ISSN={["1873-5460"]}, DOI={10.1016/j.apnum.2021.10.024}, abstractNote={Many wave propagation problems involve discontinuous material properties. We propose to solve such problems by non-overlapping domain decomposition combined with the method of difference potentials (MDP). The MDP reduces the Helmholtz equation on each subdomain to a Calderon's boundary equation with projection on the boundary. The unknowns for the Calderon's equation are the Dirichlet and Neumann data. Coupling between neighboring subdomains is rendered by applying their respective Calderon's equations to the same data at the common interface. Solutions on individual subdomains are computed concurrently using a direct solver. Our method proves to be insensitive to large jumps in the wavenumber for transmission problems, as well as interior cross-points and mixed boundary conditions, which may be a challenge to many other domain decomposition methods.}, journal={APPLIED NUMERICAL MATHEMATICS}, author={North, Evan and Tsynkov, Semyon and Turkel, Eli}, year={2022}, month={Mar}, pages={51–78} }
 @article{gilman_tsynkov_2022, title={Polarimetric radar interferometry in the presence of differential Faraday rotation}, volume={38}, ISSN={["1361-6420"]}, DOI={10.1088/1361-6420/ac5525}, abstractNote={Abstract}, number={4}, journal={INVERSE PROBLEMS}, author={Gilman, Mikhail and Tsynkov, Semyon}, year={2022}, month={Apr} }
 @article{gilman_tsynkov_2021, title={A MATHEMATICAL PERSPECTIVE ON RADAR INTERFEROMETRY}, volume={7}, ISSN={["1930-8345"]}, DOI={10.3934/ipi.2021043}, abstractNote={<p style='text-indent:20px;'>Radar interferometry is an advanced remote sensing technology that utilizes complex phases of two or more radar images of the same target taken at slightly different imaging conditions and/or different times. Its goal is to derive additional information about the target, such as elevation. While this kind of task requires centimeter-level accuracy, the interaction of radar signals with the target, as well as the lack of precision in antenna position and other disturbances, generate ambiguities in the image phase that are orders of magnitude larger than the effect of interest.</p><p style='text-indent:20px;'>Yet the common exposition of radar interferometry in the literature often skips such topics. This may lead to unrealistic requirements for the accuracy of determining the parameters of imaging geometry, unachievable precision of image co-registration, etc. To address these deficiencies, in the current work we analyze the problem of interferometric height reconstruction and provide a careful and detailed account of all the assumptions and requirements to the imaging geometry and data processing needed for a successful extraction of height information from the radar data. We employ two most popular scattering models for radar targets: an isolated point scatterer and delta-correlated extended scatterer, and highlight the similarities and differences between them.</p>}, journal={INVERSE PROBLEMS AND IMAGING}, author={Gilman, Mikhail and Tsynkov, Semyon}, year={2021}, month={Jul} }
 @article{lagergren_flores_gilman_tsynkov_2021, title={Deep Learning Approach to the Detection of Scattering Delay in Radar Images}, volume={15}, ISSN={["1559-8616"]}, DOI={10.1007/s42519-020-00149-w}, number={1}, journal={JOURNAL OF STATISTICAL THEORY AND PRACTICE}, author={Lagergren, John and Flores, Kevin and Gilman, Mikhail and Tsynkov, Semyon}, year={2021}, month={Mar} }
 @article{gilman_tsynkov_2021, title={Divergence Measures and Detection Performance for Dispersive Targets in SAR}, volume={56}, ISSN={["1944-799X"]}, DOI={10.1029/2019RS007011}, abstractNote={Abstract}, number={1}, journal={RADIO SCIENCE}, author={Gilman, Mikhail and Tsynkov, Semyon}, year={2021}, month={Jan} }
 @article{petropavlovsky_tsynkov_turkel_2021, title={Numerical Solution of 3D Unsteady Scattering Problems with Sub-linear Complexity}, DOI={10.1109/ACES53325.2021.00138}, abstractNote={We present an efficient high order accurate boundary algorithm for the numerical solution of unsteady exterior initial boundary problems for the three-dimensional wave equation. The algorithm relies on the method of difference potentials combined with the Huygens' principle.}, journal={2021 INTERNATIONAL APPLIED COMPUTATIONAL ELECTROMAGNETICS SOCIETY SYMPOSIUM (ACES)}, author={Petropavlovsky, Sergey and Tsynkov, Semyon and Turkel, Eli}, year={2021} }
 @article{medvinsky_tsynkov_turkel_2021, title={Solution of three-dimensional multiple scattering problems by the method of difference potentials}, volume={107}, ISSN={["1878-433X"]}, url={https://doi.org/10.1016/j.wavemoti.2021.102822}, DOI={10.1016/j.wavemoti.2021.102822}, abstractNote={We propose an algorithm based on the Method of Difference Potentials (MDP) for the numerical solution of multiple scattering problems in three space dimensions. The propagation of waves is assumed time-harmonic and governed by the Helmholtz equation. The latter is approximated with 6th order accuracy on a Cartesian grid by means of a compact finite difference scheme. The shape of the scatterers does not have to conform to the discretization grid, yet the MDP enables the approximation with no loss of accuracy. At the artificial outer boundary, which is spherical, the solution is terminated by a 6th order Bayliss–Gunzburger–Turkel (BGT) radiation boundary condition. The method enables efficient solution of a series of similar problems, for example, when the incident field changes while everything else stays the same, or when the type of the scattering changes (e.g., sound-soft vs. sound-hard) while the shape of the scatterer remains the same.}, journal={WAVE MOTION}, publisher={Elsevier BV}, author={Medvinsky, M. and Tsynkov, S. and Turkel, E.}, year={2021}, month={Dec} }
 @article{petropavlovsky_tsynkov_2020, title={Method of Difference Potentials for Evolution Equations with Lacunas}, volume={60}, ISSN={["1555-6662"]}, DOI={10.1134/S0965542520040144}, number={4}, journal={COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS}, author={Petropavlovsky, S. V. and Tsynkov, S. V.}, year={2020}, month={Apr}, pages={711–722} }
 @article{gilman_tsynkov_2020, title={STATISTICAL CHARACTERIZATION OF SCATTERING DELAY IN SYNTHETIC APERTURE RADAR IMAGING}, volume={14}, ISSN={["1930-8345"]}, DOI={10.3934/ipi.2020024}, abstractNote={Distinguishing between the instantaneous and delayed scatterers in synthetic aperture radar (SAR) images is important for target identification and characterization. To perform this task, one can use the autocorrelation analysis of coordinate-delay images. However, due to the range-delay ambiguity the difference in the correlation properties between the instantaneous and delayed targets may be small. Moreover, the reliability of discrimination is affected by speckle, which is ubiquitous in SAR images, and requires statistical treatment. 
Previously, we have developed a maximum likelihood based approach for discriminating between the instantaneous and delayed targets in SAR images. To test it, we employed simple statistical models. They allowed us to simulate ensembles of images that depend on various parameters, including aperture width and target contrast. 
In the current paper, we enhance our previously developed methodology by establishing confidence levels for the discrimination between the instantaneous and delayed scatterers. Our procedure takes into account the difference in thresholds for different target contrasts without making any assumptions about the statistics of those contrasts.}, number={3}, journal={INVERSE PROBLEMS AND IMAGING}, author={Gilman, Mikhail and Tsynkov, Semyon}, year={2020}, month={Jun}, pages={511–533} }
 @inproceedings{petropavlovsky_tsynkov_turkel_2019, title={A high order method of boundary operators for the 3D time-dependent wave equation}, url={https://stsynkov.math.ncsu.edu/publications/Unsteady_DPM_waves2019_v6.pdf}, booktitle={Book of Abstracts, The 14th International Conference on Mathematical
             and Numerical Aspects of Wave Propagation, WAVES 2019,
             Vienna University of Technology (TU Wien),
             Vienna, Austria, August 25-30, 2019}, author={Petropavlovsky, S. V. and Tsynkov, S. V. and Turkel, E.}, year={2019}, pages={363–365} }
 @inproceedings{petropavlovsky_tsynkov_turkel_2019, title={A method of boundary equations for unsteady hyperbolic problems in 3D}, url={https://stsynkov.math.ncsu.edu/publications/a82e_godunov90.pdf}, booktitle={Mathematics and Its Applications, Book of Abstracts for the International 
             Conference in honor of the 90th birthday of Sergei K. Godunov, 
             Novosibirsk, Russia, August 4-10, 2019}, author={Petropavlovsky, S. V. and Tsynkov, S. V. and Turkel, E.}, year={2019}, pages={68} }
 @article{smith_tsynkov_turkel_2019, title={Compact High Order Accurate Schemes for the Three Dimensional Wave Equation}, volume={5}, ISSN={0885-7474 1573-7691}, url={http://dx.doi.org/10.1007/s10915-019-00970-x}, DOI={10.1007/s10915-019-00970-x}, journal={Journal of Scientific Computing}, publisher={Springer Science and Business Media LLC}, author={Smith, F. and Tsynkov, S. and Turkel, E.}, year={2019}, month={May}, pages={1–29} }
 @article{gilman_tsynkov_2019, title={Detection of delayed target response in SAR}, volume={4}, ISSN={0266-5611 1361-6420}, url={http://dx.doi.org/10.1088/1361-6420/ab1c80}, DOI={10.1088/1361-6420/ab1c80}, abstractNote={Delayed target response in synthetic aperture radar (SAR) imaging can be obscured by the range-delay ambiguity and speckle. To analyze the range-delay ambiguity, one extends the standard SAR formulation and allows both the target reflectivity and the image to depend not only on the coordinates, but also on the response delay. However, this still leaves the speckle unaccounted for. Yet speckle is commonly found in SAR images of extended targets, and a statistical approach is usually employed to describe it. We have developed a simple model of a delayed scatterer by modifying the random function that describes a homogeneous extended scatterer. Our model allows us to obtain a relation between the deterministic parameters of the target model and statistical moments of the SAR image. We assume a regular shape of the antenna trajectory, and our model targets are localized in at least one space-time coordinate; this permits analytical formulation for statistical moments of the image. The problem of reconstruction of coordinate-delay reflectivity function is reduced to that of discrimination between instantaneous and delayed scatterers; for the latter problem, the maximum likelihood based image processing procedure has been developed. We perform Monte-Carlo simulation and evaluate performance of the classification procedure for a simple dependence of scatterer reflectivity on the delay time.}, journal={Inverse Problems}, publisher={IOP Publishing}, author={Gilman, Mikhail and Tsynkov, Semyon}, year={2019}, month={Apr}, pages={085005 (38pp)} }
 @article{medvinsky_tsynkov_turkel_2019, title={Direct implementation of high order BGT artificial boundary conditions}, volume={376}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2018.09.040}, DOI={10.1016/j.jcp.2018.09.040}, abstractNote={Local artificial boundary conditions (ABCs) for the numerical simulation of waves have been successfully used for decades (most notably, the boundary conditions due to Engquist & Majda, Bayliss, Gunzburger & Turkel, and Higdon). The basic idea behind these boundary conditions is that they cancel several leading terms in an expansion of the solution. The larger the number of terms canceled, the higher the order of the boundary condition and, in turn, the smaller the reflection error due to truncation of the original unbounded domain by an artificial outer boundary. In practice, however, the use of local ABCs has been limited to low orders (first and second), because higher order boundary conditions involve higher order derivatives of the solution, which may harm well-posedness and cause numerical instabilities. They are also difficult to implement especially in finite elements. A prominent exception is the development of local high order ABCs based on auxiliary variables. In the current paper, we implement high order Bayliss–Turkel ABCs directly — with no auxiliary variables yet no discrete approximation of the constituent high order derivatives either. Instead, we represent the solution at the boundary as an expansion with respect to a continuous basis. For the spherical artificial boundary, the basis consists of eigenfunctions of the Beltrami operator (spherical harmonics), which enable replacing the high order derivatives in the ABCs with powers of the corresponding eigenvalues. The continuous representation at the boundary is coupled to higher order compact finite differences inside the domain by the method of difference potentials (MDP). It maintains high order accuracy even when the boundary is not aligned with the discretization grid.}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Medvinsky, M. and Tsynkov, S. and Turkel, E.}, year={2019}, month={Jan}, pages={98–128} }
 @article{preface to the special issue in memory of professor saul abarbanel_2019, url={http://dx.doi.org/10.1007/s10915-019-01084-0}, DOI={10.1007/s10915-019-01084-0}, journal={Journal of Scientific Computing}, year={2019}, month={Dec} }
 @inproceedings{gilman_tsynkov_2019, title={Stochastic Models in Coordinate-Delay Synthetic Aperture Radar Imaging}, url={https://stsynkov.math.ncsu.edu/publications/Dispersive_targets_Waves2019_v3_final.pdf}, booktitle={Book of Abstracts, The 14th International Conference on Mathematical
             and Numerical Aspects of Wave Propagation, WAVES 2019,
             Vienna University of Technology (TU Wien), 
             Vienna, Austria, August 25-30, 2019}, author={Gilman, Mikhail and Tsynkov, Semyon}, year={2019}, pages={330–331} }
 @article{britt_turkel_tsynkov_2018, title={A High Order Compact Time/Space Finite Difference Scheme for the Wave Equation with Variable Speed of Sound}, volume={76}, ISSN={0885-7474 1573-7691}, url={http://dx.doi.org/10.1007/s10915-017-0639-9}, DOI={10.1007/s10915-017-0639-9}, number={2}, journal={Journal of Scientific Computing}, publisher={Springer Nature}, author={Britt, Steven and Turkel, Eli and Tsynkov, Semyon}, year={2018}, month={Jan}, pages={777–811} }
 @article{petropavlovsky_tsynkov_turkel_2018, title={A method of boundary equations for unsteady hyperbolic problems in 3D}, volume={365}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2018.03.039}, DOI={10.1016/j.jcp.2018.03.039}, abstractNote={We consider interior and exterior initial boundary value problems for the three-dimensional wave (d'Alembert) equation. First, we reduce a given problem to an equivalent operator equation with respect to unknown sources defined only at the boundary of the original domain. In doing so, the Huygens' principle enables us to obtain the operator equation in a form that involves only finite and non-increasing pre-history of the solution in time. Next, we discretize the resulting boundary equation and solve it efficiently by the method of difference potentials (MDP). The overall numerical algorithm handles boundaries of general shape using regular structured grids with no deterioration of accuracy. For long simulation times it offers sub-linear complexity with respect to the grid dimension, i.e., is asymptotically cheaper than the cost of a typical explicit scheme. In addition, our algorithm allows one to share the computational cost between multiple similar problems. On multi-processor (multi-core) platforms, it benefits from what can be considered an effective parallelization in time.}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Petropavlovsky, S. and Tsynkov, S. and Turkel, E.}, year={2018}, month={Jul}, pages={294–323} }
 @inproceedings{gilman_tsynkov_2018, title={Cross-Channel Contamination of PolSAR Images due to Frequency Dependence of Faraday Rotation Angle}, ISBN={9781538657959}, url={http://dx.doi.org/10.1109/CAMA.2018.8530603}, DOI={10.1109/CAMA.2018.8530603}, abstractNote={Compensation of the Faraday rotation (FR) effect in polarimetric synthetic aperture radar (PolSAR) involves a rotation matrix, with the FR angle determined by the magnetic field, total electron content, signal frequency, and propagation direction. We analyze the conditions where the signal frequency and/or propagation direction cannot be considered constants. In other words, the rotation matrix based on the main look direction and central radar frequency may have a significant mismatch with the received signal in fast or slow time. We derive estimates for the resulting polarimetric distortions and their effect on applications such as instrument calibration in space and measurement of the aboveground biomass.}, booktitle={2018 IEEE Conference on Antenna Measurements & Applications (CAMA)}, publisher={IEEE}, author={Gilman, Mikhail and Tsynkov, Semyon}, year={2018}, month={Sep}, pages={1–4} }
 @article{gilman_tsynkov_2018, title={Differential Faraday Rotation and Polarimetric SAR}, volume={78}, ISSN={0036-1399 1095-712X}, url={http://dx.doi.org/10.1137/17M114042X}, DOI={10.1137/17m114042x}, abstractNote={The propagation of linearly polarized electromagnetic waves through the Earth's ionosphere is accompanied by Faraday rotation (FR), which is due to gyrotropy of the ionospheric plasma in the magnet...}, number={3}, journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Gilman, Mikhail and Tsynkov, Semyon}, year={2018}, month={Jan}, pages={1422–1449} }
 @article{britt_tsynkov_turkel_2018, title={Numerical solution of the wave equation with variable wave speed on nonconforming domains by high-order difference potentials}, volume={354}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2017.10.049}, DOI={10.1016/j.jcp.2017.10.049}, abstractNote={We solve the wave equation with variable wave speed on nonconforming domains with fourth order accuracy in both space and time. This is accomplished using an implicit finite difference (FD) scheme for the wave equation and solving an elliptic (modified Helmholtz) equation at each time step with fourth order spatial accuracy by the method of difference potentials (MDP). High-order MDP utilizes compact FD schemes on regular structured grids to efficiently solve problems on nonconforming domains while maintaining the design convergence rate of the underlying FD scheme. Asymptotically, the computational complexity of high-order MDP scales the same as that for FD.}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Britt, S. and Tsynkov, S. and Turkel, E.}, year={2018}, month={Feb}, pages={26–42} }
 @inproceedings{petropavlovsky_tsynkov_turkel_2017, title={An efficient numerical algorithm for the 3D wave equation in domains
             of complex shape}, url={https://stsynkov.math.ncsu.edu/publications/wavesPetropTsynkov_v3.pdf}, booktitle={Mathematical
             and Numerical Aspects of Wave Propagation WAVES 2017,
             The 13th International Conference, Minneapolis, MN, USA, May 15--19, 2017.
             Book of Abstracts}, author={Petropavlovsky, S. and Tsynkov, S. and Turkel, E.}, year={2017}, pages={365–366} }
 @inproceedings{osintcev_tsynkov_2017, title={Computational complexity of artificial boundary conditions
             for Maxwell's equations in the FDTD method}, url={https://stsynkov.math.ncsu.edu/publications/OsTsy_4.pdf}, booktitle={Mathematical
             and Numerical Aspects of Wave Propagation WAVES 2017,
             The 13th International Conference, Minneapolis, MN, USA, May 15--19, 2017.
             Book of Abstracts}, author={Osintcev, Mikhail and Tsynkov, Semyon}, year={2017}, pages={275–276} }
 @article{gilman_smith_tsynkov_gilman_smith_tsynkov_2017, title={Conventional SAR imaging}, ISBN={["978-3-319-52125-1"]}, DOI={10.1007/978-3-319-52127-5_2}, abstractNote={In this chapter, we explain the fundamental principles of SAR data collection and image formation, i.e., inversion of the received data. Synthetic aperture radar uses microwaves for imaging the surface of the Earth from airplanes or satellites. Unlike photography which generates the picture by essentially recoding the intensity of the light reflected off the different parts of the target, SAR imaging exploits the phase information of the interrogating signals and as such can be categorized as a coherent imaging technology.}, journal={TRANSIONOSPHERIC SYNTHETIC APERTURE IMAGING}, author={Gilman, Mikhail and Smith, Erick and Tsynkov, Semyon and Gilman, M and Smith, E and Tsynkov, S}, year={2017}, pages={19–57} }
 @inproceedings{britt_tsynkov_turkel_2017, title={High order accurate solution of the wave equation by
             compact finite differences and difference potentials}, url={https://stsynkov.math.ncsu.edu/publications/waves2017_Steven_v4.pdf}, booktitle={Mathematical
             and Numerical Aspects of Wave Propagation WAVES 2017,
             The 13th International Conference, Minneapolis, MN, USA, May 15--19, 2017.
             Book of Abstracts}, author={Britt, Steven and Tsynkov, Semyon and Turkel, Eli}, year={2017}, pages={63–64} }
 @inproceedings{magura_petropavlovsky_tsynkov_turkel_2017, title={High order numerical solution of the Helmholtz
             equation for domains with reentrant corners}, url={https://stsynkov.math.ncsu.edu/publications/waves2017_Steven_v4.pdf}, booktitle={Mathematical
             and Numerical Aspects of Wave Propagation WAVES 2017,
             The 13th International Conference, Minneapolis, MN, USA, May 15--19, 2017.
             Book of Abstracts}, author={Magura, S. and Petropavlovsky, S. and Tsynkov, S. and Turkel, E.}, year={2017}, pages={367–368} }
 @article{magura_petropavlovsky_tsynkov_turkel_2017, title={High-order numerical solution of the Helmholtz equation for domains with reentrant corners}, volume={118}, ISSN={0168-9274}, url={http://dx.doi.org/10.1016/j.apnum.2017.02.013}, DOI={10.1016/j.apnum.2017.02.013}, abstractNote={Standard numerical methods often fail to solve the Helmholtz equation accurately near reentrant corners, since the solution may become singular. The singularity has an inhomogeneous contribution from the boundary data near the corner and a homogeneous contribution that is determined by boundary conditions far from the corner. We present a regularization algorithm that uses a combination of analytical and numerical tools to distinguish between these two contributions and ultimately subtract the singularity. We then employ the method of difference potentials to numerically solve the regularized problem with high-order accuracy over a domain with a curvilinear boundary. Our numerical experiments show that the regularization successfully restores the design rate of convergence.}, journal={Applied Numerical Mathematics}, publisher={Elsevier BV}, author={Magura, S. and Petropavlovsky, S. and Tsynkov, S. and Turkel, E.}, year={2017}, month={Aug}, pages={87–116} }
 @article{gilman_smith_tsynkov_2017, title={Inverse scattering off anisotropic targets}, journal={Transionospheric synthetic aperture imaging}, author={Gilman, M. and Smith, E. and Tsynkov, S.}, year={2017}, pages={373–415} }
 @inproceedings{gilman_tsynkov_2017, title={Mathematical analysis of SAR imaging through a turbulent ionosphere}, volume={1895}, ISSN={["0094-243X"]}, url={http://dx.doi.org/10.1063/1.5007357}, DOI={10.1063/1.5007357}, abstractNote={Synthetic aperture radar (SAR) imaging though the Earth ionosphere is subject to distortions due to ionospheric turbulence. We consider the limiting cases of small-scale and large-scale turbulence and characterize the distortions in terms of image blurring and azimuthal shift. It is shown that in the large-scale case, a high level of eikonal fluctuations can coexist with the low degree of image distortions, and that blurring becomes significant at much higher levels of fluctuations than the shift. In the small-scale case, a low level of eikonal fluctuations is a precondition for imaging, while the magnitude of distortions depends on the ratio between the eikonal correlation radius and the length of the synthetic aperture.}, publisher={Author(s)}, author={Gilman, M. and Tsynkov, S.}, editor={Todorov, Michail D.Editor}, year={2017} }
 @article{gilman_smith_tsynkov_gilman_smith_tsynkov_2017, title={Modeling radar targets beyond the first Born approximation}, ISBN={["978-3-319-52125-1"]}, DOI={10.1007/978-3-319-52127-5_7}, abstractNote={In this chapter, we return to the foundations of the SAR ambiguity theory SAR ambiguity theory that we presented in Chapter 2 , and address the inconsistencies of the conventional approach outlined in Section 2.7 A standard representation of the image in the SAR ambiguity theory is SAR image by the convolution integral convolution ( 2.1 ) [see also ( 2.31 )]:}, journal={TRANSIONOSPHERIC SYNTHETIC APERTURE IMAGING}, author={Gilman, Mikhail and Smith, Erick and Tsynkov, Semyon and Gilman, M and Smith, E and Tsynkov, S}, year={2017}, pages={311–371} }
 @article{petropavlovsky_tsynkov_2017, title={Non-deteriorating time domain numerical algorithms for Maxwell's electrodynamics}, volume={336}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2017.01.068}, DOI={10.1016/j.jcp.2017.01.068}, abstractNote={The Huygens' principle and lacunae can help construct efficient far-field closures for the numerical simulation of unsteady waves propagating over unbounded regions. Those closures can be either standalone or combined with other techniques for the treatment of artificial outer boundaries. A standalone lacunae-based closure can be thought of as a special artificial boundary condition (ABC) that is provably free from any error associated with the domain truncation. If combined with a different type of ABC or a perfectly matched layer (PML), a lacunae-based approach can help remove any long-time deterioration (e.g., instability) that arises at the outer boundary regardless of why it occurs in the first place. A specific difficulty associated with Maxwell's equations of electromagnetism is that in general their solutions do not have classical lacunae and rather have quasi-lacunae. Unlike in the classical case, the field inside the quasi-lacunae is not zero; instead, there is an electrostatic solution driven by the electric charges that accumulate over time. In our previous work [23], we have shown that quasi-lacunae can also be used for building the far-field closures. However, for achieving a provably non-deteriorating performance over arbitrarily long time intervals, the accumulated charges need to be known ahead of time. The main contribution of the current paper is that we remove this limitation and modify the algorithm in such a way that one can rather avoid the accumulation of charge all together. Accordingly, the field inside the quasi-lacunae becomes equal to zero, which facilitates obtaining the temporally uniform error estimates as in the case of classical lacunae. The performance of the modified algorithm is corroborated by a series of numerical simulations. The range of problems that the new method can address includes important combined formulations, for which the interior subproblem may be non-Huygens', and only the exterior subproblem, i.e., the far field, is Huygens'.}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Petropavlovsky, S. and Tsynkov, S.}, year={2017}, month={May}, pages={1–35} }
 @article{gilman_smith_tsynkov_gilman_smith_tsynkov_2017, title={SAR imaging through the Earth's ionosphere}, ISBN={["978-3-319-52125-1"]}, DOI={10.1007/978-3-319-52127-5_3}, abstractNote={When the signal of a spaceborne radar travels between the satellite and the ground, it becomes subject to the temporal dispersion of radio waves in the Earth’s ionosphere [18]. dispersion temporal The dispersion distorts the signal, and if the matched filter does not properly account for that, a mismatch occurs and the quality of the image deteriorates. The extent of deterioration becomes smaller as the ratio of the Langmuir frequency of the ionospheric plasma to the carrier frequency of the radar decreases. This is a part of the reason why many modern spaceborne SAR instruments operate in higher frequency bands. For example, TerraSAR-X operates in the X-band, on the frequency of 9.6GHz. X-band}, journal={TRANSIONOSPHERIC SYNTHETIC APERTURE IMAGING}, author={Gilman, Mikhail and Smith, Erick and Tsynkov, Semyon and Gilman, M and Smith, E and Tsynkov, S}, year={2017}, pages={59–161} }
 @inproceedings{gilman_tsynkov_2017, title={The Doppler Effect for SAR}, url={https://stsynkov.math.ncsu.edu/publications/waves_2017_Doppler_v3.pdf}, booktitle={Mathematical
             and Numerical Aspects of Wave Propagation WAVES 2017,
             The 13th International Conference, Minneapolis, MN, USA, May 15--19, 2017.
             Book of Abstracts}, author={Gilman, Mikhail and Tsynkov, Semyon}, year={2017}, pages={369–370} }
 @article{gilman_smith_tsynkov_gilman_smith_tsynkov_2017, title={The effect of ionospheric anisotropy}, ISBN={["978-3-319-52125-1"]}, DOI={10.1007/978-3-319-52127-5_5}, abstractNote={In Chapter 3, we have shown that the Earth’s ionosphere exerts an adverse effect on SAR imaging. It is due to the mismatch between the actual radar signal affected by the dispersion of radio waves in the ionosphere and the matched filter used for signal processing. Accordingly, to improve the image one should correct the filter. This requires knowledge of the total electron content in the ionosphere, as well as of another parameter that characterizes the azimuthal variation of the electron number density (see Section 3.9). These quantities can be reconstructed by probing the ionosphere on two distinct carrier frequencies and exploiting the resulting redundancy in the data (see Section 3.10).}, journal={TRANSIONOSPHERIC SYNTHETIC APERTURE IMAGING}, author={Gilman, Mikhail and Smith, Erick and Tsynkov, Semyon and Gilman, M and Smith, E and Tsynkov, S}, year={2017}, pages={217–264} }
 @article{gilman_smith_tsynkov_gilman_smith_tsynkov_2017, title={The effect of ionospheric turbulence}, ISBN={["978-3-319-52125-1"]}, DOI={10.1007/978-3-319-52127-5_4}, abstractNote={In Chapter 3, we have shown that temporal dispersion of the propagation medium (Earth’s ionosphere) causes distortions of SAR images (see Section 3.8). Moreover, we have identified the key integral characteristics of the ionospheric plasma that allow one to quantify those distortions. They are the zeroth moment of the electron number density N e, i.e., the TEC N H given by (3.66), as well the first moment $$\mathcal{Q}$$ of the azimuthal derivative of N e defined by ( 3.182 ). We have also demonstrated that one can obtain the unknown quantities N H and $$\mathcal{Q}$$ with the help of dual carrier probing (see Section 3.10 ) and subsequently incorporate them into the SAR matched filter matched filter in order to effectively eliminate the distortions (see Section 3.11 ). This correction of the filter is possible because one and the same pair of values $$(N_{H},\mathcal{Q})$$ “serves” all antenna signals used for the construction of the image, i.e., all the terms in the azimuthal sum. Once the values of N H and $$\mathcal{Q}$$ have been derived, the corrected filter will match the received signals for all antenna positions along the synthetic array.}, journal={TRANSIONOSPHERIC SYNTHETIC APERTURE IMAGING}, author={Gilman, Mikhail and Smith, Erick and Tsynkov, Semyon and Gilman, M and Smith, E and Tsynkov, S}, year={2017}, pages={163–215} }
 @article{gilman_smith_tsynkov_gilman_smith_tsynkov_2017, title={The start-stop approximation}, ISBN={["978-3-319-52125-1"]}, DOI={10.1007/978-3-319-52127-5_6}, abstractNote={For the analysis of the SAR data inversion algorithm in Chapters 2 through 5, we have employed the start-stop approximation, which is considered standard in the literature, see, e.g., [25, 40, 76, 79] and also [86]. It assumes that the radar antenna is at standstill while it sends the interrogating pulse toward the target and receives the scattered response, after which the antenna moves down the flight track to the position where the next pulse is emitted and received.}, journal={TRANSIONOSPHERIC SYNTHETIC APERTURE IMAGING}, author={Gilman, Mikhail and Smith, Erick and Tsynkov, Semyon and Gilman, M and Smith, E and Tsynkov, S}, year={2017}, pages={265–309} }
 @book{gilman_smith_tsynkov_gilman_smith_tsynkov_2017, place={Cham, Switzerland}, title={Transionospheric Synthetic Aperture Imaging}, ISBN={9783319521251 9783319521275}, ISSN={2296-5009 2296-5017}, url={http://dx.doi.org/10.1007/978-3-319-52127-5}, DOI={10.1007/978-3-319-52127-5}, abstractNote={This landmark monograph presents the most recent mathematical developments in the analysis of ionospheric distortions of SAR images and offers innovative new strategies for their mitigation. As a prer}, journal={Applied and Numerical Harmonic Analysis}, publisher={Springer International Publishing}, author={Gilman, M. and Smith, E. and Tsynkov, S. and Gilman, M and Smith, E and Tsynkov, S}, year={2017}, pages={1–1} }
 @article{gilman_smith_tsynkov_2017, title={Transionospheric Synthetic Aperture Imaging Discussion and outstanding questions}, journal={Transionospheric synthetic aperture imaging}, author={Gilman, M. and Smith, E. and Tsynkov, S.}, year={2017}, pages={417–431} }
 @article{gilman_smith_tsynkov_2017, title={Transionospheric synthetic aperture imaging Introduction}, journal={Transionospheric synthetic aperture imaging}, author={Gilman, M. and Smith, E. and Tsynkov, S.}, year={2017}, pages={1–17} }
 @inproceedings{fedoseyev_kansa_tsynkov_petropavlovskiy_osintcev_shumlak_henshaw_2016, title={A universal framework for non-deteriorating time-domain numerical algorithms in Maxwell’s electrodynamics}, volume={1773}, ISSN={["0094-243X"]}, url={http://dx.doi.org/10.1063/1.4964955}, DOI={10.1063/1.4964955}, abstractNote={We present the implementation of the Lacuna method, that removes a key diffculty that currently hampers many existing methods for computing unsteady electromagnetic waves on unbounded regions. Numerical accuracy and/or stability may deterio-rate over long times due to the treatment of artificial outer boundaries. We describe a developed universal algorithm and software that correct this problem by employing the Huygens’ principle and lacunae of Maxwell’s equations.The algorithm provides a temporally uniform guaranteed error bound (no deterioration at all), and the software will enable robust electromagnetic simulations in a high-performance computing environment. The methodology applies to any geometry, any scheme, and any boundary condition. It eliminates the long-time deterioration regardless of its origin and how it manifests itself. In retrospect, the lacunae method was first proposed by V. Ryaben’kii and subsequently developed by S. Tsynkov.We have completed development of an innovative numerical met...}, publisher={Author(s)}, author={Fedoseyev, A. and Kansa, E. J. and Tsynkov, S. and Petropavlovskiy, S. and Osintcev, M. and Shumlak, U. and Henshaw, W. D.}, editor={Todorov, Michail D.Editor}, year={2016} }
 @inbook{britt_petropavlovsky_tsynkov_turkel_2016, title={Difference Potentials Methods for Hyperbolic Problems Using High Order Finite Difference Schemes}, url={https://stsynkov.math.ncsu.edu/publications/icosahomAbstract2016.pdf}, booktitle={Book of Abstracts, International Conference on Spectral and High Order Methods, ICOSAHOM 2016, 
             Rio De Janeiro, Brazil, June 2016}, author={Britt, Steven and Petropavlovsky, Sergei and Tsynkov, Semyon and Turkel, Eli}, year={2016} }
 @article{medvinsky_tsynkov_turkel_2016, title={Solving the Helmholtz equation for general smooth geometry using simple grids}, volume={62}, ISSN={0165-2125}, url={http://dx.doi.org/10.1016/j.wavemoti.2015.12.004}, DOI={10.1016/j.wavemoti.2015.12.004}, abstractNote={The method of difference potentials was originally proposed by Ryaben’kii, and is a generalized discrete version of the method of Calderon’s operators. It handles non-conforming curvilinear boundaries, variable coefficients, and non-standard boundary conditions while keeping the complexity of the solver at the level of a finite-difference scheme on a regular structured grid. Compact finite difference schemes enable high order accuracy on small stencils and so require no additional boundary conditions beyond those needed for the differential equation itself. Previously, we have used difference potentials combined with compact schemes for solving transmission/scattering problems in regions of a simple shape. In this paper, we generalize our previous work to incorporate smooth general shaped boundaries and interfaces, including a formulation that involves multiple scattering.}, journal={Wave Motion}, publisher={Elsevier BV}, author={Medvinsky, M. and Tsynkov, S. and Turkel, E.}, year={2016}, month={Apr}, pages={75–97} }
 @article{gilman_tsynkov_2015, title={A Mathematical Model for SAR Imaging beyond the First Born Approximation}, volume={8}, ISSN={1936-4954}, url={http://dx.doi.org/10.1137/140973025}, DOI={10.1137/140973025}, abstractNote={The assumption of weak scattering is standard for the mathematical analysis of synthetic aperture radar (SAR), as it helps linearize the inverse problem via the first Born approximation and thus makes it amenable to solution. Yet it is not consistent with another common assumption, that the interrogating waves do not penetrate into the target material and get scattered off its surface only, which essentially means that the scattering is strong. In the paper, we revisit the foundations of the SAR ambiguity theory in order to address this and other existing inconsistencies, such as the absence of the Bragg scale in scattering. We introduce a new model for radar targets that allows us to compute the scattered field from first principles. This renders the assumption of weak scattering unnecessary yet keeps the overall inverse scattering problem linear. Finally, we show how one can incorporate the Leontovich boundary condition into SAR ambiguity theory.}, number={1}, journal={SIAM Journal on Imaging Sciences}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Gilman, Mikhail and Tsynkov, Semyon}, year={2015}, month={Jan}, pages={186–225} }
 @article{britt_petropavlovsky_tsynkov_turkel_2015, title={Computation of singular solutions to the Helmholtz equation with high order accuracy}, volume={93}, ISSN={0168-9274}, url={http://dx.doi.org/10.1016/j.apnum.2014.10.006}, DOI={10.1016/j.apnum.2014.10.006}, abstractNote={Solutions to elliptic PDEs, in particular to the Helmholtz equation, become singular near the boundary if the boundary data do not possess sufficient regularity. In that case, the convergence of standard numerical approximations may slow down or cease altogether. We propose a method that maintains a high order of grid convergence even in the presence of singularities. This is accomplished by an asymptotic expansion that removes the singularities up to several leading orders, and the remaining regularized part of the solution can then be computed on the grid with the expected accuracy. The computation on the grid is rendered by a compact finite difference scheme combined with the method of difference potentials. The scheme enables fourth order accuracy on a narrow 3×3 stencil: it uses only one unknown variable per grid node and requires only as many boundary conditions as needed for the underlying differential equation itself. The method of difference potentials enables treatment of non-conforming boundaries on regular structured grids with no deterioration of accuracy, while the computational complexity remains comparable to that of a conventional finite difference scheme on the same grid. The method of difference potentials can be considered a generalization of the method of Calderon's operators in PDE theory. In the paper, we provide a theoretical analysis of our combined methodology and demonstrate its numerical performance on a series of tests that involve Dirichlet and Neumann boundary data with various degrees of “non-regularity”: an actual jump discontinuity, a discontinuity in the first derivative, a discontinuity in the second derivative, etc. All computations are performed on a Cartesian grid, whereas the boundary of the domain is a circle, chosen as a simple but non-conforming shape. In all cases, the proposed methodology restores the design rate of grid convergence, which is fourth order, in spite of the singularities and regardless of the fact that the boundary is not aligned with the discretization grid. Moreover, as long as the location of the singularities is known and remains fixed, a broad spectrum of problems involving different boundary conditions and/or data on “smooth” segments of the boundary can be solved economically since the discrete counterparts of Calderon's projections need to be calculated only once and then can be applied to each individual formulation at very little additional cost.}, journal={Applied Numerical Mathematics}, publisher={Elsevier BV}, author={Britt, S. and Petropavlovsky, S. and Tsynkov, S. and Turkel, E.}, year={2015}, month={Jul}, pages={215–241} }
 @article{epshteyn_sofronov_tsynkov_2015, title={Professor V.S. Ryaben'kii. On the occasion of the 90-th birthday}, volume={93}, ISSN={0168-9274}, url={http://dx.doi.org/10.1016/j.apnum.2015.02.001}, DOI={10.1016/j.apnum.2015.02.001}, abstractNote={Numerical approximations and computational modeling of problems from Biology and Materials Science often deal with partial differential equations with varying coefficients and domains with irregular geometry. The challenge here is to design an efficient and accurate numerical method that can resolve properties of solutions in different domains/subdomains, while handling the arbitrary geometries of the domains. In this work, we consider 2D elliptic models with material interfaces and develop efficient high-order accurate methods based on Difference Potentials for such problems.}, journal={Applied Numerical Mathematics}, publisher={Elsevier BV}, author={Epshteyn, Yekaterina and Sofronov, Ivan and Tsynkov, Semyon}, year={2015}, month={Jul}, pages={1–2} }
 @inbook{medvinsky_turkel_tsynkov_2015, title={Transmission and Scattering of Waves by General Shapes with High Order Accuracy
             Using the Difference Potentials Method}, url={https://stsynkov.math.ncsu.edu/publications/Waves2015-book-of-abstracts.pdf}, booktitle={Book of Abstracts, The 12th International Conference on
             Mathematical and Numerical Aspects of Wave Propagation, WAVES 2015,
             Karlsruhe, Germany,  July 20--24, 2015}, author={Medvinsky, M. and Turkel, E. and Tsynkov, S.}, year={2015}, pages={256–257} }
 @article{godunov_zhukov_lazarev_sofronov_turchaninov_kholodov_tsynkov_chetverushkin_epshteyn_2015, title={Viktor Solomonovich Ryaben'kii and his school (on his 90th birthday)}, volume={70}, ISSN={0036-0279 1468-4829}, url={http://dx.doi.org/10.1070/RM2015V070N06ABEH004981}, DOI={10.1070/RM2015V070N06ABEH004981}, abstractNote={Professor Viktor Solomonovich Ryaben’kii, doctor of the physical and mathematical sciences and a prominent expert in computational mathematics, observed his 90th birthday on 20 March 2013. He was born in Moscow, in the family of state employees Solomon Abramovich Ryaben’kii and Berta Pavlovna Ryaben’kaya. In 1940 he enrolled in the Faculty of Mechanics and Mathematics of Moscow State University, but World War II interrupted his studies. After many narrow escapes from death, the mechanic and driver Ryaben’kii greeted Victory Day as a sergeant in the Guards, returned to his dear Faculty, graduated in 1949, and was admitted for graduate studies. The talents of the future leading figure of Russian computational mathematics were recognized and supported by Ivan Georgievich Petrovsky, the prominent mathematician and administrator of the sciences, who supervised Ryaben’kii’s work first for his diploma thesis and then for his Ph.D. dissertation. This dissertation, under the long and— for that time — highly specialized title “Stability of finite-difference schemes and the application of the method of finite differences to solution of the Cauchy problem for systems of partial differential equations”, defended at Moscow State University in 1952, revealed a new name to the computation community, and over time justified inscribing Ryaben’kii’s name in the list of illustrious founders of the theory of finite-difference schemes. It so happened that the young specialist Ryaben’kii began his career at about the same time as the (then secret) Department of Applied Mathematics was formed, initially as a division of the Mathematical Institute. Mstislav Vsevoldovich Keldysh, a brilliant mathematician, an expert in mechanics, and a statesman (who subsequently became President of the Academy of Sciences (1961–1975) at the age of 50) was its head. The Department of Applied Mathematics was organized to solve computational problems in nuclear and thermonuclear power generation,}, number={6}, journal={Russian Mathematical Surveys}, publisher={IOP Publishing}, author={Godunov, S K and Zhukov, V T and Lazarev, M I and Sofronov, I L and Turchaninov, V I and Kholodov, A S and Tsynkov, S V and Chetverushkin, B N and Epshteyn, Ye Yu}, year={2015}, month={Dec}, pages={1183–1210} }
 @article{годунов_godunov_жуков_zhukov_лазарев_lazarev_софронов_sofronov_турчанинов_turchaninov_et al._2015, title={Виктор Соломонович Рябенький и его школа (к девяностолетию со дня рождения)}, volume={70}, ISSN={0042-1316 2305-2872}, url={http://dx.doi.org/10.4213/rm9676}, DOI={10.4213/rm9676}, note={[in Russian]}, number={6(426)}, journal={Uspekhi Matematicheskikh Nauk}, publisher={Steklov Mathematical Institute}, author={Годунов, Сергей Константинович and Godunov, Sergei Konstantinovich and Жуков, Виктор Тимофеевич and Zhukov, Victor Timofeevich and Лазарев, М И and Lazarev, M I and Софронов, Иван Львович and Sofronov, Ivan L'vovich and Турчанинов, Виктор Игоревич and Turchaninov, Viktor Igorevich and et al.}, year={2015}, pages={213–236} }
 @inproceedings{gilman_tsynkov_2014, place={Berlin-Offenbach, Germany}, title={Detection of Material Dispersion Using SAR}, url={https://stsynkov.math.ncsu.edu/publications/eusar-2014.pdf}, booktitle={Proceedings of the 10th European Conference on Synthetic Aperture Radar (EUSAR 2014)}, publisher={VDE VERLAG GMBH}, author={Gilman, Mikhail and Tsynkov, Semyon}, year={2014}, pages={1013–1016} }
 @article{gilman_smith_tsynkov_2014, title={Single-polarization SAR imaging in the presence of Faraday rotation}, volume={30}, ISSN={0266-5611 1361-6420}, url={http://dx.doi.org/10.1088/0266-5611/30/7/075002}, DOI={10.1088/0266-5611/30/7/075002}, abstractNote={We discuss the single-polarization SAR imaging with the Faraday rotation (FR) taken into account. The FR leads to a reduction in the intensity of the received radar signal that varies over the signal length. That, in turn, results in a degradation of the image. In particular, the image of a point target may have its intensity peak split in the range direction. To distinguish between the cases of low reflectivity and those where the low antenna signal is due to the FR, we employ the image autocorrelation analysis. This analysis also helps determine the parameters of the FR, which, in turn, allow us to introduce an approach for correcting the single-polarization SAR images distorted by FR.}, number={7}, journal={Inverse Problems}, publisher={IOP Publishing}, author={Gilman, Mikhail and Smith, Erick and Tsynkov, Semyon}, year={2014}, month={Jun}, pages={075002} }
 @article{britt_tsynkov_turkel_2013, title={A High-Order Numerical Method for the Helmholtz Equation with Nonstandard Boundary Conditions}, volume={35}, ISSN={1064-8275 1095-7197}, url={http://dx.doi.org/10.1137/120902689}, DOI={10.1137/120902689}, abstractNote={We describe a high-order accurate methodology for the numerical simulation of time-harmonic waves governed by the Helmholtz equation. Our approach combines compact finite difference schemes that provide an inexpensive venue toward high-order accuracy with the method of difference potentials developed by Ryaben'kii. The latter can be interpreted as a generalized discrete version of the method of Calderon's operators in the theory of partial differential equations. The method of difference potentials can accommodate nonconforming boundaries on regular structured grids with no loss of accuracy due to staircasing. It introduces a universal framework for treating boundary conditions of any type. A significant advantage of this method is that changing the boundary condition within a fairly broad variety does not require any major changes to the algorithm and is computationally inexpensive. In this paper, we address various types of boundary conditions using the method of difference potentials. We demonstrate th...}, number={5}, journal={SIAM Journal on Scientific Computing}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Britt, D. S. and Tsynkov, S. V. and Turkel, E.}, year={2013}, month={Jan}, pages={A2255–A2292} }
 @article{turkel_gordon_gordon_tsynkov_2013, title={Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number}, volume={232}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2012.08.016}, DOI={10.1016/j.jcp.2012.08.016}, abstractNote={Several studies have presented compact fourth order accurate finite difference approximation for the Helmholtz equation in two or three dimensions. Several of these formulae allow for the wave number k to be variable. Other papers have extended this further to include variable coefficients within the Laplacian which models non-homogeneous materials in electromagnetism. Later papers considered more accurate compact sixth order methods but these were restricted to constant k. In this paper we extend these compact sixth order schemes to variable k in both two and three dimensions. Results on 2D and 3D problems with known analytic solutions verify the sixth order accuracy. We demonstrate that for large wave numbers, the second order scheme cannot produce comparable results with reasonable grid sizes.}, number={1}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Turkel, Eli and Gordon, Dan and Gordon, Rachel and Tsynkov, Semyon}, year={2013}, month={Jan}, pages={272–287} }
 @article{kansa_shumlak_tsynkov_2013, title={Discrete Calderon’s projections on parallelepipeds and their application to computing exterior magnetic fields for FRC plasmas}, volume={234}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2012.09.033}, DOI={10.1016/j.jcp.2012.09.033}, abstractNote={Confining dense plasma in a field reversed configuration (FRC) is considered a promising approach to fusion. Numerical simulation of this process requires setting artificial boundary conditions (ABCs) for the magnetic field because whereas the plasma itself occupies a bounded region (within the FRC coils), the field extends from this region all the way to infinity. If the plasma is modeled using single fluid magnetohydrodynamics (MHD), then the exterior magnetic field can be considered quasi-static. This field has a scalar potential governed by the Laplace equation. The quasi-static ABC for the magnetic field is obtained using the method of difference potentials, in the form of a discrete Calderon boundary equation with projection on the artificial boundary shaped as a parallelepiped. The Calderon projection itself is computed by convolution with the discrete fundamental solution on the three-dimensional Cartesian grid.}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Kansa, E. and Shumlak, U. and Tsynkov, S.}, year={2013}, month={Feb}, pages={172–198} }
 @inbook{britt_medvinsky_turkel_tsynkov_2013, title={High Order Numerical Simulation of the Transmission and 
             Scattering of Waves Using the Method of Difference Potentials}, url={https://stsynkov.math.ncsu.edu/publications/Tsynkov_abstract_for_Ryabenkii-90.pdf}, booktitle={Proceedings of the International Conference Difference Schemes and Applications 
             in honor of the 90-th Birthday of Prof. V. S. Ryaben'kii, Moscow, Russia, May 27--31, 2013}, author={Britt, S. and Medvinsky, M. and Turkel, E. and Tsynkov, S.}, year={2013}, pages={33–34} }
 @article{medvinsky_tsynkov_turkel_2013, title={High order numerical simulation of the transmission and scattering of waves using the method of difference potentials}, volume={243}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2013.03.014}, DOI={10.1016/j.jcp.2013.03.014}, abstractNote={The method of difference potentials generalizes the method of Calderon’s operators from PDEs to arbitrary difference equations and systems. It offers several key advantages, such as the capability of handling boundaries/interfaces that are not aligned with the discretization grid, variable coefficients, and nonstandard boundary conditions. In doing so, the complexity of the algorithm remains comparable to that of an ordinary finite difference scheme on a regular structured grid. Previously, we have applied the method of difference potentials to solving several variable coefficient interior Helmholtz problems with fourth and sixth order accuracy. We have employed compact finite difference schemes as a core discretization methodology. Those schemes enable high order accuracy on narrow stencils and hence require only as many boundary conditions as needed for the underlying differential equation itself. Numerical experiments corroborate the high order accuracy of our method for variable coefficients, regular grids, and non-conforming boundaries. In the current paper, we extend the previously developed methodology to exterior problems. We present a complete theoretical analysis of the algorithm, as well as the results of a series of numerical simulations. Specifically, we study the scattering of time-harmonic waves about smooth shapes, subject to various boundary conditions. We also solve the transmission/scattering problems, in which not only do the waves scatter off a given shape but also propagate through the interface and travel across the heterogeneous medium inside. In all the cases, our methodology guarantees high order accuracy for regular grids and non-conforming boundaries and interfaces.}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Medvinsky, M. and Tsynkov, S. and Turkel, E.}, year={2013}, month={Jun}, pages={305–322} }
 @article{gilman_smith_tsynkov_2013, title={Reduction of ionospheric distortions for spaceborne synthetic aperture radar with the help of image registration}, volume={29}, ISSN={0266-5611 1361-6420}, url={http://dx.doi.org/10.1088/0266-5611/29/5/054005}, DOI={10.1088/0266-5611/29/5/054005}, abstractNote={We propose a robust technique for reducing the ionospheric distortions in spaceborne synthetic aperture radar (SAR) images. It is based on probing the terrain on two distinct carrier frequencies. Compared to our previous work on the subject (Smith and Tsynkov 2011 SIAM J. Imaging Sciences 4 501–42), the increase in robustness is achieved by applying an area-based image registration algorithm to the two images obtained on two frequencies. This enables an accurate evaluation of the shift between the two images, which, in turn, translates into an accurate estimate of the total electron content and its along-track gradient in the ionosphere. These estimates allow one to correct the matched filter and thus improve the quality of the image. Moreover, for the analysis of SAR resolution in the current paper we take into account the Ohm conductivity in the ionosphere (in addition to its temporal dispersion), and also consider the true Kolmogorov spectrum of the ionospheric turbulence, as opposed to its approximate representation that we have used previously.}, number={5}, journal={Inverse Problems}, publisher={IOP Publishing}, author={Gilman, Mikhail and Smith, Erick and Tsynkov, Semyon}, year={2013}, month={Apr}, pages={054005} }
 @article{gilman_smith_tsynkov_2012, title={A linearized inverse scattering problem for the polarized waves and anisotropic targets}, volume={28}, ISSN={0266-5611 1361-6420}, url={http://dx.doi.org/10.1088/0266-5611/28/8/085009}, DOI={10.1088/0266-5611/28/8/085009}, abstractNote={We analyze the scattering of a plane transverse linearly polarized electromagnetic wave off a plane interface between the vacuum and a given material. For a variety of predominantly dielectric materials, from isotropic to anisotropic and weakly conductive, we show that when the scattering is weak, the first Born approximation predicts the correct scattered field in the vacuum region. We also formulate and solve the corresponding linearized inverse scattering problem. Specifically, we provide a necessary and sufficient condition under which interpreting the target material as a weakly conductive uniaxial crystal allows one to reconstruct all the degrees of freedom contained in the complex 2 × 2 Sinclair scattering matrix. This development can help construct a full-fledged radar ambiguity theory for polarimetric imaging by means of a synthetic aperture radar (SAR), which is in contrast to the approach that currently dominates the SAR literature and exploits a fully phenomenological scattering matrix. Moreover, the linearized scattering off a material half-space naturally gives rise to the ground reflectivity function in the form of a single layer (i.e. a δ-layer) at the interface. A ground reflectivity function of this type is often introduced in the SAR literature without a rigorous justification. Besides the conventional SAR analysis, we expect that the proposed approach may appear useful for the material identification SAR (miSAR) purposes.}, number={8}, journal={Inverse Problems}, publisher={IOP Publishing}, author={Gilman, Mikhail and Smith, Erick and Tsynkov, Semyon}, year={2012}, month={Jul}, pages={085009} }
 @article{petropavlovsky_tsynkov_2012, title={A non-deteriorating algorithm for computational electromagnetism based on quasi-lacunae of Maxwell’s equations}, volume={231}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2011.09.019}, DOI={10.1016/j.jcp.2011.09.019}, abstractNote={The performance of many well-known methods used for the treatment of outer boundaries in computational electromagnetism (CEM) may deteriorate over long time intervals. The methods found susceptible to this undesirable phenomenon include some local low order artificial boundary conditions (ABCs), as well as perfectly matched layers (PMLs). We propose a universal algorithm for correcting this problem. It works regardless of either why the deterioration occurs in each particular instance, or how it actually manifests itself (loss of accuracy, loss of stability, etc.). Our algorithm relies on the Huygens’ principle in the generalized form, when a non-zero electrostatic solution can be present behind aft fronts of the propagating waves, i.e., inside the lacunae of Maxwell’s equations. In this case, we refer to quasi-lacunae as opposed to conventional lacunae, for which the solution behind aft fronts is zero. The use of quasi-lacunae allows us to overcome a key constraint of the previously developed version of our algorithm that was based on genuine lacunae. Namely, the currents that drive the solution no longer have to be solenoidal. Another important development is that we apply the methodology to general non-Huygens’ problems.}, number={2}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Petropavlovsky, S.V. and Tsynkov, S.V.}, year={2012}, month={Jan}, pages={558–585} }
 @article{medvinsky_tsynkov_turkel_2012, title={Erratum to: The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes}, volume={53}, ISSN={0885-7474 1573-7691}, url={http://dx.doi.org/10.1007/S10915-012-9638-Z}, DOI={10.1007/S10915-012-9638-Z}, number={2}, journal={Journal of Scientific Computing}, publisher={Springer Science and Business Media LLC}, author={Medvinsky, M. and Tsynkov, S. and Turkel, E.}, year={2012}, month={Sep}, pages={482–482} }
 @article{medvinsky_tsynkov_turkel_2012, title={The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes}, volume={53}, ISSN={0885-7474 1573-7691}, url={http://dx.doi.org/10.1007/s10915-012-9602-y}, DOI={10.1007/s10915-012-9602-y}, number={1}, journal={Journal of Scientific Computing}, publisher={Springer Science and Business Media LLC}, author={Medvinsky, M. and Tsynkov, S. and Turkel, E.}, year={2012}, month={May}, pages={150–193} }
 @article{smith_tsynkov_2011, title={Dual Carrier Probing for Spaceborne SAR Imaging}, volume={4}, ISSN={1936-4954}, url={http://dx.doi.org/10.1137/10078325X}, DOI={10.1137/10078325x}, abstractNote={Spaceborne imaging of the Earth's surface by synthetic aperture radar (SAR) may be adversely affected by the ionosphere that causes distortions of the signals emitted and received by the radar antenna. In our previous publication on the subject [SIAM J. Imaging Sci., 2 (2009) pp. 140-182], we have analyzed those distortions for the inhomogeneous ionosphere described by the cold plasma model. Based on the analysis, we have concluded that the deterioration of SAR images was due to the mismatch between certain parameters of the actual received signal, which is slowed down by the temporal dispersion in the ionosphere, and the corresponding parameters of the matched filter, which is taken as if the propagation were unobstructed. Consequently, to improve the quality of the images, the filter must be corrected. However, to get the appropriate correction, one needs to know some key characteristics of the ionosphere precisely at the time and place the image is taken. To obtain those characteristics, we currently propose probing the terrain, and hence the ionosphere, on two distinct carrier frequencies. We also investigate the performance of the matched filters that were corrected this way and show that the final quality of the images, i.e., their resolution and sharpness evaluated using the SAR ambiguity theory, indeed improves.}, number={2}, journal={SIAM Journal on Imaging Sciences}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Smith, E. M. and Tsynkov, S. V.}, year={2011}, month={Jan}, pages={501–542} }
 @inbook{turkel_tsynkov_2011, title={Interfaces for the Helmholtz Equation with High Order Accuracy}, url={https://stsynkov.math.ncsu.edu/publications/a66e_waves11.pdf}, booktitle={Proceedings of the 10th International Conference on
             Mathematical and Numerical Aspects of Wave Propagation, WAVES 2011,
             Vancouver, Canada, July 25--29, 2011}, author={Turkel, E. and Tsynkov, S.}, year={2011}, pages={659–662} }
 @article{britt_tsynkov_turkel_2011, title={Numerical Simulation of Time-Harmonic Waves in Inhomogeneous Media using Compact High Order Schemes}, volume={9}, ISSN={1815-2406 1991-7120}, url={http://dx.doi.org/10.4208/cicp.091209.080410s}, DOI={10.4208/cicp.091209.080410s}, abstractNote={Abstract}, number={3}, journal={Communications in Computational Physics}, publisher={Global Science Press}, author={Britt, Steven and Tsynkov, Semyon and Turkel, Eli}, year={2011}, month={Mar}, pages={520–541} }
 @article{petropavlovsky_tsynkov_2011, title={Quasi-Lacunae of Maxwell's Equations}, volume={71}, ISSN={0036-1399 1095-712X}, url={http://dx.doi.org/10.1137/100798041}, DOI={10.1137/100798041}, abstractNote={Classical lacunae in the solutions of hyperbolic differential equations and systems (in the spaces of odd dimension) are a manifestation of the Huygens' principle. If the source terms are compactly supported in space and time, then, at any finite location in space, the solution becomes identically zero after a finite interval of time. In other words, the propagating waves have sharp aft fronts. For Maxwell's equations though, even if the currents that drive the field are compactly supported in time, they may still lead to the accumulation of charges. In that case, the solution won't have the lacunae per se. We show, however, that the notion of classical lacunae can be generalized, and that even when the steady-state charges are present, the waves still have sharp aft fronts. Yet behind those aft fronts, there is a nonzero electrostatic solution rather than one identically zero.}, number={4}, journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Petropavlovsky, S. V. and Tsynkov, S. V.}, year={2011}, month={Jan}, pages={1109–1122} }
 @article{britt_tsynkov_turkel_2010, title={A Compact Fourth Order Scheme for the Helmholtz Equation in Polar Coordinates}, volume={45}, ISSN={0885-7474 1573-7691}, url={http://dx.doi.org/10.1007/s10915-010-9348-3}, DOI={10.1007/s10915-010-9348-3}, number={1-3}, journal={Journal of Scientific Computing}, publisher={Springer Science and Business Media LLC}, author={Britt, S. and Tsynkov, S. and Turkel, E.}, year={2010}, month={Jan}, pages={26–47} }
 @article{baruch_fibich_tsynkov_2009, title={A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media}, volume={228}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2009.02.014}, DOI={10.1016/j.jcp.2009.02.014}, abstractNote={We present a novel computational methodology for solving the scalar nonlinear Helmholtz equation (NLH) that governs the propagation of laser light in Kerr dielectrics. The methodology addresses two well-known challenges in nonlinear optics: Singular behavior of solutions when the scattering in the medium is assumed predominantly forward (paraxial regime), and the presence of discontinuities in the optical properties of the medium. Specifically, we consider a slab of nonlinear material which may be grated in the direction of propagation and which is immersed in a linear medium as a whole. The key components of the methodology are a semi-compact high-order finite-difference scheme that maintains accuracy across the discontinuities and enables sub-wavelength resolution on large domains at a tolerable cost, a nonlocal two-way artificial boundary condition (ABC) that simultaneously facilitates the reflectionless propagation of the outgoing waves and forward propagation of the given incoming waves, and a nonlinear solver based on Newton’s method. The proposed methodology combines and substantially extends the capabilities of our previous techniques built for 1D and for multi-D. It facilitates a direct numerical study of nonparaxial propagation and goes well beyond the approaches in the literature based on the “augmented” paraxial models. In particular, it provides the first ever evidence that the singularity of the solution indeed disappears in the scalar NLH model that includes the nonparaxial effects. It also enables simulation of the wavelength-width spatial solitons, as well as of the counter-propagating solitons.}, number={10}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Baruch, G. and Fibich, G. and Tsynkov, S.}, year={2009}, month={Jun}, pages={3789–3815} }
 @article{ryaben’kii_tsynkov_utyuzhnikov_2009, title={Active control of sound with variable degree of cancellation}, volume={22}, ISSN={0893-9659}, url={http://dx.doi.org/10.1016/j.aml.2009.07.010}, DOI={10.1016/j.aml.2009.07.010}, abstractNote={We formulate and solve a control problem for the field (e.g., time-harmonic sound) governed by a linear PDE or system on a composite domain in Rn. Namely, we require that simultaneously and independently on each subdomain the sound generated in its complement be attenuated to a desired degree. This goal is achieved by adding special control sources defined only at the interface between the subdomains. We present a general solution for controls in the continuous and discrete setting.}, number={12}, journal={Applied Mathematics Letters}, publisher={Elsevier BV}, author={Ryaben’kii, V.S. and Tsynkov, S.V. and Utyuzhnikov, S.V.}, year={2009}, month={Dec}, pages={1846–1851} }
 @article{ryaben’kii_utyuzhnikov_tsynkov_2009, title={Difference Problem of Noise Suppression and Other Problems of Active Control for Time-Harmonic Sound over Composite Regions}, volume={425}, number={4}, journal={Doklady Rossiiskoi Akademii Nauk, Matematika (Transactions of the Russian Academy of Sciences, Mathematics)}, author={Ryaben’kii, V.S. and Utyuzhnikov, S.V. and Tsynkov, S.V.}, year={2009}, pages={456–458} }
 @article{ryaben’kii_utyuzhnikov_tsynkov_2009, title={Difference problem of noise suppression and other problems of active control of single-frequency sound on a composite domain}, volume={79}, ISSN={1064-5624 1531-8362}, url={http://dx.doi.org/10.1134/S1064562409020240}, DOI={10.1134/S1064562409020240}, note={[Russian]}, number={2}, journal={Doklady Mathematics}, publisher={Pleiades Publishing Ltd}, author={Ryaben’kii, V. S. and Utyuzhnikov, S. V. and Tsynkov, S. V.}, year={2009}, month={Apr}, pages={240–242} }
 @article{lim_utyuzhnikov_lam_turan_avis_ryaben'kii_tsynkov_2009, title={Experimental Validation of the Active Noise Control Methodology Based on Difference Potentials}, volume={47}, ISSN={0001-1452 1533-385X}, url={http://dx.doi.org/10.2514/1.32496}, DOI={10.2514/1.32496}, abstractNote={To achieve active noise cancellation over a large area, it is often necessary to get a measure of the physical properties of the noise source to devise a counter measure. This, however, is not practical in many cases. A mathematical approach, the Difference Potential Method, can provide an alternative solution for active shielding over a large area. In this approach, the cancellation of unwanted noise requires only measurements near the boundary surface but not at the source itself, and it does not require any other information. Moreover, the solution based on difference potentials applies to bounded domains in the presence of acoustic sources inside the domain to be shielded. This paper reports on the results of experimental validation. It has been demonstrated that while preserving the wanted sound, the developed approach can cancel out the unwanted noise. The volumetric noise cancellation offered by the proposed methodology along with leaving the wanted sound unchanged is a unique feature compared to other techniques available in the literature. It can be most useful in the context of applications related to civil aviation, in particular, for eliminating the exterior noise inside the passenger compartments of both current and future generation of commercial aircraft.}, number={4}, journal={AIAA Journal}, publisher={American Institute of Aeronautics and Astronautics (AIAA)}, author={Lim, H. and Utyuzhnikov, S. V. and Lam, Y. W. and Turan, A. and Avis, M. R. and Ryaben'kii, V. S. and Tsynkov, S. V.}, year={2009}, month={Apr}, pages={874–884} }
 @article{baruch_fibich_tsynkov_turkel_2009, title={Fourth order schemes for time-harmonic wave equations with
              discontinuous coefficients}, volume={5}, url={http://global-sci.org/intro/article_detail/cicp/7742.html}, number={2-4}, journal={Commun. Comput. Phys.}, author={Baruch, Guy and Fibich, Gadi and Tsynkov, Semyon and Turkel, Eli}, year={2009}, pages={442–455} }
 @inproceedings{baruch_fibich_tsynkov_turkel_2009, title={Fourth order schemes for time-harmonic wave equations with discontinuous coefficients}, volume={5}, number={2-4}, booktitle={Communications in Computational Physics}, author={Baruch, G. and Fibich, G. and Tsynkov, S. and Turkel, E.}, year={2009}, pages={442–455} }
 @article{abarbanel_qasimov_tsynkov_2009, title={Long-Time Performance of Unsplit PMLs with Explicit Second Order Schemes}, volume={41}, ISSN={0885-7474 1573-7691}, url={http://dx.doi.org/10.1007/s10915-009-9282-4}, DOI={10.1007/s10915-009-9282-4}, abstractNote={A gradual long-time growth of the solution in perfectly matched layers (PMLs) has been previously reported in the literature. This undesirable phenomenon may hamper the performance of the layer, which is designed to truncate the computational domain for unsteady wave propagation problems. For unsplit PMLs, prior studies have attributed the growth to the presence of multiple eigenvalues in the amplification matrix of the governing system of differential equations. In the current paper, we analyze the temporal behavior of unsplit PMLs for some commonly used second order explicit finite-difference schemes that approximate the Maxwell’s equations. Our conclusion is that on top of having the PML as a potential source of long-time growth, the type of the layer and the choice of the scheme play a major role in how rapidly this growth may manifest itself and whether or not it manifests itself at all.}, number={1}, journal={Journal of Scientific Computing}, publisher={Springer Nature}, author={Abarbanel, S. and Qasimov, H. and Tsynkov, S.}, year={2009}, month={Mar}, pages={1–12} }
 @inbook{baruch_fibich_tsynkov_2009, title={Numerical Simulation of Focusing Nonlinear Waves in the Nonparaxial Regime}, url={https://stsynkov.math.ncsu.edu/publications/a54e_waves09-abstract.pdf}, booktitle={Proceedings of the 9th International Conference on
             Mathematical and Numerical Aspects of Wave Propagation, WAVES 2009,
             Pau, France, June 15--19, 2009}, author={Baruch, G. and Fibich, G. and Tsynkov, S. V.}, editor={Barucq, Helene and Bonnet-Bendhia, Anne-Sophie and Cohen, Gary and Diaz, Julien and Ezziana, Abdelaaziz and Joly, PatrickEditors}, year={2009}, pages={325–327} }
 @inbook{baruch_fibich_tsynkov_2009, title={Numerical solution of the nonlinear Helmholtz equation}, url={https://stsynkov.math.ncsu.edu/publications/a54e_godunov80.pdf}, note={[in Russian]}, booktitle={Mathematics in Applications, Proceedings of the conference in honor of the 80th birthday
             of Academician S. K. Godunov, Novosibirsk, Russia, July 20--24, 2009}, author={Baruch, G. and Fibich, G. and Tsynkov, S. V.}, year={2009}, pages={37–38} }
 @article{tsynkov_2009, title={On SAR Imaging through the Earth's Ionosphere}, volume={2}, ISSN={1936-4954}, url={http://dx.doi.org/10.1137/080721509}, DOI={10.1137/080721509}, abstractNote={We analyze the effect of dispersion of radio waves in the Earth's ionosphere on the performance (image resolution) of spaceborne synthetic aperture radars (SARs). We describe the electromagnetic propagation in the framework of a scalar model for the transverse field subject to weak anomalous dispersion due to the cold plasma. Random contributions to the refraction index are accounted for by the Kolmogorov model of ionospheric turbulence. A key consideration used when analyzing the statistics of waves is normalization of the probability distributions for long propagation distances. The ionospheric phenomena, both deterministic and random, are shown to affect the azimuthal resolution of a SAR sensor stronger than the range resolution; also, the effect of randomness appears weaker than that of the baseline dispersion. Specific quantitative estimates are provided for some typical values of the key parameters. Probing on two carrier frequencies is identified as a possible venue for reducing the ionospheric distortions.}, number={1}, journal={SIAM Journal on Imaging Sciences}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Tsynkov, S. V.}, year={2009}, month={Jan}, pages={140–182} }
 @inbook{tsynkov_2009, title={On SAR Imaging through the Earth's Ionosphere}, url={https://stsynkov.math.ncsu.edu/publications/a52e_waves09-abstract.pdf}, booktitle={Proceedings of the 9th International Conference on
             Mathematical and Numerical Aspects of Wave Propagation, WAVES 2009,
             Pau, France, June 15--19, 2009}, author={Tsynkov, S. V.}, editor={Barucq, Helene and Bonnet-Bendhia, Anne-Sophie and Cohen, Gary and Diaz, Julien and Ezziana, Abdelaaziz and Joly, PatrickEditors}, year={2009}, pages={312–313} }
 @article{tsynkov_2009, title={On the Use of Start-Stop Approximation for Spaceborne SAR Imaging}, volume={2}, ISSN={1936-4954}, url={http://dx.doi.org/10.1137/08074026X}, DOI={10.1137/08074026x}, abstractNote={The start-stop approximation is a standard tool for processing radar data in synthetic aperture imaging. It assumes that the antenna is motionless when a pulse is emitted and the scattered signal received, after which the antenna moves to its next sending/receiving position along the flight track. However, when the antenna is mounted on a satellite, as opposed to an airplane, its relatively high speed raises at least two questions. The first one is whether the image may be affected by the actual displacement of the antenna during the pulse round-trip time between the orbit and the Earth's surface. This displacement, in fact, can be rather large. Nonetheless, by analyzing the corresponding generalized ambiguity function of a synthetic aperture radar (SAR) sensor we show that in practice this issue can be disregarded. The second question is related to the Doppler frequency shift, which, again, is larger for spaceborne radars than for airborne radars. In the early SAR studies, this frequency shift provided a venue for understanding the azimuthal resolution of a radar. However, in a more rigorous analysis based on the generalized ambiguity function, the Doppler effect is typically left out of consideration. We show that for the image to stay largely unaffected by Doppler, the frequency shift must be included in the definition of a matched filter. Otherwise, there will be a geometric shift (translation) of the entire imaged scene from its true position, and there may also be a slight deterioration of the image sharpness (contrast).}, number={2}, journal={SIAM Journal on Imaging Sciences}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Tsynkov, S. V.}, year={2009}, month={Jan}, pages={646–669} }
 @article{qasimov_tsynkov_2008, title={Lacunae based stabilization of PMLs}, volume={227}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2008.04.018}, DOI={10.1016/j.jcp.2008.04.018}, abstractNote={Perfectly matched layers (PMLs) are used for the numerical solution of wave propagation problems on unbounded regions. They surround the finite computational domain (obtained by truncation) and are designed to attenuate and completely absorb all the outgoing waves while producing no reflections from the interface between the domain and the layer. PMLs have demonstrated excellent performance for many applications. However, they have also been found prone to instabilities that manifest themselves when the simulation time is long. Hereafter, we propose a modification that stabilizes any PML applied to a hyperbolic partial differential equation/system that satisfies the Huygens’ principle (such as the 3D d’Alembert equation or Maxwell’s equations in vacuum). The modification makes use of the presence of lacunae in the corresponding solutions and allows us to establish a temporally uniform error bound for arbitrarily long-time intervals. At the same time, it does not change the original PML equations. Hence, the matching properties of the layer, as well as any other properties deemed important, are fully preserved. We also emphasize that besides the aforementioned PML instabilities per se, the methodology can be used to cure any other undesirable long-term computational phenomenon, such as the accuracy loss of low order absorbing boundary conditions.}, number={15}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Qasimov, H. and Tsynkov, S.}, year={2008}, month={Jul}, pages={7322–7345} }
 @inbook{fibich_tsynkov_2008, title={Numerical Solution of the Nonlinear Helmholtz Equation}, volume={5}, ISBN={9781584885689 9781420010879}, ISSN={2154-7386}, url={http://dx.doi.org/10.1201/9781420010879.ch2}, DOI={10.1201/9781420010879.ch2}, abstractNote={Нелинейное уравнение Гельмгольца описывает распространение интенсивных пучков лазерного излучения в (прозрачных) диэлектриках типа Керра, когда основным явлением, представляющим интерес с точки зрения физики, является самофокусировка. В докладе будет описан разностный метод высокого порядка точности для решения этого уравнения, позволяющий проводить расчеты в том числе и для сред с разрывными оптическими характеристиками. Метод использует компактные разности четвертого порядка, включая односторонние компактные разности для аппроксимации условий на поверхностях разрыва. Ключевыми элементами метода являются высокоточные нелокальные искусственные краевые условия, обеспечивающие прозрачность внешних границ для всех исходящих волн и одновременно задающие входящие лазерные пучки, а также способ решения получающиейся системы разностных уравнений, основанный на линеаризации по Ньютону и требующий специального подхода из-за того, что нелинейность типа Керра недифференцируема для комплекснозначных решений. Предлагаемый метод эффективен для исследования важного и долго остававшегося нерешенным вопроса в нелинейной оптике, а именно вопроса о возникновении особенности решения в случае, когда рассеяние света в среде предполагается направленным преимущественно вперед (так называемое параксиальное приближение). Впервые удалось получить решения без особенности для тех режимов, для которых решение в параксиальном приближении, которое описывается нелинейным уравнением Шредингера, перестает существовать уже на конеч-}, booktitle={Effective Computational Methods for Wave Propagation}, publisher={Chapman and Hall/CRC}, author={Fibich, G and Tsynkov, S}, year={2008}, month={Feb}, pages={37–62} }
 @article{baruch_fibich_tsynkov_2008, title={Simulations of the nonlinear Helmholtz equation: arrest of beam collapse, nonparaxial solitons and counter-propagating beams}, volume={16}, ISSN={1094-4087}, url={http://dx.doi.org/10.1364/OE.16.013323}, DOI={10.1364/OE.16.013323}, abstractNote={We solve the (2+1)D nonlinear Helmholtz equation (NLH) for input beams that collapse in the simpler NLS model. Thereby, we provide the first ever numerical evidence that nonparaxiality and backscattering can arrest the collapse. We also solve the (1+1)D NLH and show that solitons with radius of only half the wavelength can propagate over forty diffraction lengths with no distortions. In both cases we calculate the backscattered field, which has not been done previously. Finally, we compute the dynamics of counter-propagating solitons using the NLH model, which is more comprehensive than the previously used coupled NLS model.}, number={17}, journal={Optics Express}, publisher={The Optical Society}, author={Baruch, G. and Fibich, G. and Tsynkov, Semyon}, year={2008}, month={Aug}, pages={13323} }
 @article{peterson_tsynkov_2007, title={Active Control of Sound for Composite Regions}, volume={67}, ISSN={0036-1399 1095-712X}, url={http://dx.doi.org/10.1137/060662368}, DOI={10.1137/060662368}, abstractNote={We present a methodology for the active control of time-harmonic wave fields, e.g., acoustic disturbances, in composite regions. This methodology extends our previous approach developed for the case of arcwise connected regions. The overall objective is to eliminate the effect of all outside field sources on a given domain of interest, i.e., to shield this domain. In this context, active shielding means introducing additional field sources, called active controls, that generate the annihilating signal and cancel out the unwanted component of the field. As such, the problem of active shielding can be interpreted as a special inverse source problem for the governing differential equation or system. For a composite domain, not only do the controls prevent interference from all exterior sources, but they can also enforce a predetermined communication pattern between the individual subdomains (as many as desired). In other words, they either allow the subdomains to communicate freely with one another or otherw...}, number={6}, journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Peterson, A. W. and Tsynkov, S. V.}, year={2007}, month={Jan}, pages={1582–1609} }
 @inbook{baruch_fibich_tsynkov_2007, title={High-Order Numerical Method for the Nonlinear Helmholtz Equation 
             with Material Discontinuities}, url={https://stsynkov.math.ncsu.edu/publications/Waves_2007_proceedings.pdf}, booktitle={Proceedings of the 8th International Conference on
             Mathematical and Numerical Aspects of Wave Propagation, WAVES 2007,
             University of Reading, UK, July 23 -- 27, 2007}, author={Baruch, G. and Fibich, G. and Tsynkov, S. V.}, editor={Biggs, Nick and Bonnet-Bendhia, Anne-Sophie and Chamberlain, Peter and Chandler-Wildea, Simon and Cohen, Gary and Haddar, Houssem and Joly, Patrick and Langdon, Stephen and Lunéville, Eric and Pelloni, Beatrice and et al.Editors}, year={2007}, pages={455–457} }
 @article{baruch_fibich_tsynkov_2007, title={High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension}, volume={227}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2007.08.022}, DOI={10.1016/j.jcp.2007.08.022}, abstractNote={The nonlinear Helmholtz equation (NLH) models the propagation of electromagnetic waves in Kerr media, and describes a range of important phenomena in nonlinear optics and in other areas. In our previous work, we developed a fourth order method for its numerical solution that involved an iterative solver based on freezing the nonlinearity. The method enabled a direct simulation of nonlinear self-focusing in the nonparaxial regime, and a quantitative prediction of backscattering. However, our simulations showed that there is a threshold value for the magnitude of the nonlinearity, above which the iterations diverge. In this study, we numerically solve the one-dimensional NLH using a Newton-type nonlinear solver. Because the Kerr nonlinearity contains absolute values of the field, the NLH has to be recast as a system of two real equations in order to apply Newton’s method. Our numerical simulations show that Newton’s method converges rapidly and, in contradistinction with the iterations based on freezing the nonlinearity, enables computations for very high levels of nonlinearity. In addition, we introduce a novel compact finite-volume fourth order discretization for the NLH with material discontinuities. Our computations corroborate the design fourth order convergence of the method. The one-dimensional results of the current paper create a foundation for the analysis of multidimensional problems in the future.}, number={1}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Baruch, G. and Fibich, G. and Tsynkov, S.}, year={2007}, month={Nov}, pages={820–850} }
 @article{baruch_fibich_tsynkov_2007, title={High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry}, volume={204}, ISSN={0377-0427}, url={http://dx.doi.org/10.1016/j.cam.2006.01.048}, DOI={10.1016/j.cam.2006.01.048}, abstractNote={The nonlinear Helmholtz (NLH) equation models the propagation of intense laser beams in a Kerr medium. The NLH takes into account the effects of nonparaxiality and backward scattering that are neglected in the more common nonlinear Schrödinger model. In [G. Fibich, S. Tsynkov, High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering, J. Comput. Phys., 171 (2001) 632–677] and [G. Fibich, S. Tsynkov, Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions, J. Comput. Phys., 210 (2005) 183–224], a novel high-order numerical method for solving the NLH was introduced and implemented in the case of a two-dimensional Cartesian geometry. The NLH was solved iteratively, using the separation of variables and a special nonlocal two-way artificial boundary condition applied to the resulting decoupled linear systems. In the current paper, we propose a major improvement to the previous method. Instead of using LU decomposition after the separation of variables, we employ an efficient summation rule that evaluates convolution with the discrete Green's function. We also extend the method to a three-dimensional setting with cylindrical symmetry, under both Dirichlet and Sommerfeld-type transverse boundary conditions.}, number={2}, journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier BV}, author={Baruch, G. and Fibich, G. and Tsynkov, S.}, year={2007}, month={Jul}, pages={477–492} }
 @article{ryaben’kii_tsynkov_utyuzhnikov_2007, title={Inverse source problem and active shielding for composite domains}, volume={20}, ISSN={0893-9659}, url={http://dx.doi.org/10.1016/j.aml.2006.05.019}, DOI={10.1016/j.aml.2006.05.019}, abstractNote={The problem of active shielding (AS) for a multiply connected domain consists of constructing additional sources of the field (e.g., acoustic) so that all individual subdomains can either communicate freely with one another or otherwise be shielded from their peers. This problem can be interpreted as a special inverse source problem for the differential equation (or system) that governs the field. In the paper, we obtain general solution for a discretized composite AS problem and show that it reduces to solving a collection of auxiliary problems for simply connected domains.}, number={5}, journal={Applied Mathematics Letters}, publisher={Elsevier BV}, author={Ryaben’kii, V.S. and Tsynkov, S.V. and Utyuzhnikov, S.V.}, year={2007}, month={May}, pages={511–515} }
 @inbook{qasimov_tsynkov_2007, title={Lacuna-based stabilization of PMLs}, url={https://stsynkov.math.ncsu.edu/publications/Waves_2007_proceedings.pdf}, booktitle={Proceedings of the 8th International Conference on
             Mathematical and Numerical Aspects of Wave Propagation, WAVES 2007,
             University of Reading, UK, July 23 -- 27, 2007}, author={Qasimov, H. and Tsynkov, S.}, editor={Biggs, Nick and Bonnet-Bendhia, Anne-Sophie and Chamberlain, Peter and Chandler-Wildea, Simon and Cohen, Gary and Haddar, Houssem and Joly, Patrick and Langdon, Stephen and Lunéville, Eric and Pelloni, Beatrice and et al.Editors}, year={2007}, pages={298–300} }
 @book{kurganov_tsynkov_2007, place={Raleigh, NC}, title={On Spectral Accuracy of Quadrature Formulae Based on
                   Piecewise Polynomial Interpolation}, url={https://stsynkov.math.ncsu.edu/publications/a43e5.crsc.pdf}, number={CRSC-TR07-11}, institution={Center for Research in Scientific Computation, North Carolina State University}, author={Kurganov, A. and Tsynkov, S.}, year={2007} }
 @article{tsynkov_2007, title={Weak Lacunae of Electromagnetic Waves in Dilute Plasma}, volume={67}, ISSN={0036-1399 1095-712X}, url={http://dx.doi.org/10.1137/060655134}, DOI={10.1137/060655134}, abstractNote={The propagation of waves is said to be diffusionless, and the corresponding governing PDE (or system) is said to satisfy Huygens' principle if the waves due to compactly supported sources have sharp aft fronts. The areas of no disturbance behind the aft fronts are called lacunae. Diffusionless propagation of waves is rare, whereas its opposite—diffusive propagation accompanied by aftereffects—is common. Nonetheless, lacunae can still be observed in a number of important applications, including the Maxwell equations in vacuum or in dielectrics with static response. In the framework of these applications, lacunae can be efficiently exploited for the numerical simulation of unsteady waves, and considerable progress has been made toward the development of lacunae-based methods for computational electromagnetism. Maxwell equations in vacuum are Huygens' because they reduce to a set of d'Alembert equations. Besides d'Alembert equations, there are no other scalar Huygens' equations in the standard $3+1$-dimensio...}, number={6}, journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Tsynkov, S. V.}, year={2007}, month={Jan}, pages={1548–1581} }
 @book{ryaben’kii_tsynkov_2006, place={Boca Raton, FL}, title={A Theoretical Introduction to Numerical Analysis}, ISBN={9781420011166}, url={http://dx.doi.org/10.1201/9781420011166}, DOI={10.1201/9781420011166}, abstractNote={PREFACE ACKNOWLEDGMENTS INTRODUCTION Discretization Conditioning Error On Methods of Computation INTERPOLATION OF FUNCTIONS. QUADRATURES ALGEBRAIC INTERPOLATION Existence and Uniqueness of Interpolating Polynomial Classical Piecewise Polynomial Interpolation Smooth Piecewise Polynomial Interpolation (Splines) Interpolation of Functions of Two Variables TRIGONOMETRIC INTERPOLATION Interpolation of Periodic Functions Interpolation of Functions on an Interval. Relation between Algebraic and Trigonometric Interpolation COMPUTATION OF DEFINITE INTEGRALS. QUADRATURES Trapezoidal Rule, Simpson's Formula, and the Like Quadrature Formulae with No Saturation. Gaussian Quadratures Improper Integrals. Combination of Numerical and Analytical Methods Multiple Integrals SYSTEMS OF SCALAR EQUATIONS SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS: DIRECT METHODS Different Forms of Consistent Linear Systems Linear Spaces, Norms, and Operators Conditioning of Linear Systems Gaussian Elimination and Its Tri-Diagonal Version Minimization of Quadratic Functions and Its Relation to Linear Systems The Method of Conjugate Gradients Finite Fourier Series ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS Richardson Iterations and the Like Chebyshev Iterations and Conjugate Gradients Krylov Subspace Iterations Multigrid Iterations OVERDETERMINED LINEAR SYSTEMS. THE METHOD OF LEAST SQUARES Examples of Problems that Result in Overdetermined Systems Weak Solutions of Full Rank Systems. QR Factorization Rank Deficient Systems. Singular Value Decomposition NUMERICAL SOLUTION OF NONLINEAR EQUATIONS AND SYSTEMS Commonly Used Methods of Rootfinding Fixed Point Iterations Newton's Method THE METHOD OF FINITE DIFFERENCES FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS NUMERCAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS Examples of Finite-Difference Schemes. Convergence Approximation of Continuous Problem by a Difference Scheme. Consistency Stability of Finite-Difference Schemes The Runge-Kutta Methods Solution of Boundary Value Problems Saturation of Finite-Difference Methods The Notion of Spectral Methods FINITE-DIFFERENCE SCHEMES FOR PARTIAL DIFFERENTIAL EQUATIONS Key Definitions and Illustrating Examples Construction of Consistent Difference Schemes Spectral Stability Criterion for Finite-Difference Cauchy Problems Stability for Problems with Variable Coefficients Stability for Initial Boundary Value Problems Explicit and Implicit Schemes for the Heat Equation DISCONTINUOUS SOLUTIONS AND METHODS OF THEIR COMPUTATION Differential Form of an Integral Conservation Law Construction of Difference Schemes DISCRETE METHODS FOR ELLIPTIC PROBLEMS A Simple Finite-Difference Scheme. The Maximum Principle The Notion of Finite Elements. Ritz and Galerkin Approximations THE METHODS OF BOUNDARY EQUATIONS FOR THE NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS BOUNDARY INTEGRAL EQUATIONS AND THE METHOD OF BOUNDARY ELEMENTS Reduction of Boundary Value Problems to Integral Equations Discretization of Integral Equations and Boundary Elements The Range of Applicability for Boundary Elements BOUNDARY EQUATIONS WITH PROJECTIONS AND THE METHOD OF DIFFERENCE POTENTIALS Formulation of Model Problems Difference Potentials Solution of Model Problems LIST OF FIGURES REFERENCED BOOKS REFERENCED JOURNAL ARTICLES INDEX}, publisher={Chapman and Hall/CRC}, author={Ryaben’kii, V.S. and Tsynkov, S.V.}, year={2006}, month={Nov}, pages={xiv+537} }
 @article{ryaben’kii_utyuzhnikov_tsynkov_2006, title={The Problem of Active Shielding for Composite Regions}, volume={411}, url={https://stsynkov.math.ncsu.edu/publications/a45r.pdf}, note={[Russian]}, number={2}, journal={Dokl. Akad. Nauk}, author={Ryaben’kii, V. S. and Utyuzhnikov, S. V. and Tsynkov, S. V.}, year={2006}, pages={164–166} }
 @article{ryaben’kii_utyuzhnikov_tsynkov_2006, title={The Problem of Active Shielding for Multiply Connected Regions}, volume={411}, number={2}, journal={Doklady Rossiiskoi Akademii Nauk Matematika (Transactions of the Russian Academy of Sciences, Mathematics)}, author={Ryaben’kii, V.S. and Utyuzhnikov, S.V. and Tsynkov, S.V.}, year={2006}, pages={164–166} }
 @article{ryaben'kii_utyuzhnikov_tsynkov_2006, title={The problem of active noise shielding in composite domains}, volume={74}, DOI={10.1134/S106456240606007X}, number={3}, journal={Doklady. Mathematics}, author={Ryaben'kii, V. S. and Utyuzhnikov, S. V. and Tsynkov, Semyon}, year={2006}, pages={812–814} }
 @inproceedings{fibich_tsynkov_2005, title={Numerical Solution of the Nonlinear Helmholtz Equation Using
             Nonorthogonal Expansions}, booktitle={Proceedings of the 7th International Conference on
             Mathematical and Numerical Aspects of Wave Propagation, WAVES 2005,
             Brown University, Providence, RI, June 20 -- 24, 2005}, author={Fibich, G. and Tsynkov, S. V.}, year={2005}, pages={379–381} }
 @article{fibich_tsynkov_2005, title={Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions}, volume={210}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2005.04.015}, DOI={10.1016/j.jcp.2005.04.015}, abstractNote={In [J. Comput. Phys. 171 (2001) 632–677] we developed a fourth-order numerical method for solving the nonlinear Helmholtz equation which governs the propagation of time-harmonic laser beams in media with a Kerr-type nonlinearity. A key element of the algorithm was a new nonlocal two-way artificial boundary condition (ABC), set in the direction of beam propagation. This two-way ABC provided for reflectionless propagation of the outgoing waves while also fully transmitting the given incoming beam at the boundaries of the computational domain. Altogether, the algorithm of [J. Comput. Phys. 171 (2001) 632–677] has allowed for a direct simulation of nonlinear self-focusing without neglecting nonparaxial effects and backscattering. To the best of our knowledge, this capacity has never been achieved previously in nonlinear optics. In the current paper, we propose an improved version of the algorithm. The principal innovation is that instead of using the Dirichlet boundary conditions in the direction orthogonal to that of the laser beam propagation, we now introduce Sommerfeld-type local radiation boundary conditions, which are constructed directly in the discrete framework. Numerically, implementation of the Sommerfeld conditions requires evaluation of eigenvalues and eigenvectors for a non-Hermitian matrix. Subsequently, the separation of variables, which is a key building block of the aforementioned nonlocal ABC, is implemented through an expansion with respect to the nonorthogonal basis of the eigenvectors. Numerical simulations show that the new algorithm offers a considerable improvement in its numerical performance, as well as in the range of physical phenomena that it is capable of simulating.}, number={1}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Fibich, G. and Tsynkov, S.}, year={2005}, month={Nov}, pages={183–224} }
 @article{lončarić_tsynkov_2005, title={Quadratic optimization in the problems of active control of sound}, volume={52}, ISSN={0168-9274}, url={http://dx.doi.org/10.1016/j.apnum.2004.08.041}, DOI={10.1016/j.apnum.2004.08.041}, abstractNote={We analyze the problem of suppressing the unwanted component of a time-harmonic acoustic field (noise) on a predetermined region of interest. The suppression is rendered by active means, i.e., by introducing the additional acoustic sources called controls that generate the appropriate anti-sound. Previously, we have obtained general solutions for active controls in both continuous and discrete formulation of the problem. We have also obtained optimal solutions that minimize the overall absolute acoustic source strength of active control sources, which is equivalent to minimization in the sense of L1. By contrast, in the current paper we formulate and study optimization problems that involve quadratic functions of merit. Specifically, we minimize the L2 norm of the control sources, and we consider both the unconstrained and constrained minimization. The unconstrained L2 minimization is an easy problem to address numerically. On the other hand, the constrained approach allows one to analyze sophisticated geometries. In a special case, we can compare our finite-difference optimal solutions to the continuous optimal solutions obtained previously using a semi-analytic technique. We also show that the optima obtained in the sense of L2 differ drastically from those obtained in the sense of L1.}, number={4}, journal={Applied Numerical Mathematics}, publisher={Elsevier BV}, author={Lončarić, J. and Tsynkov, S.V.}, year={2005}, month={Mar}, pages={381–400} }
 @article{tsynkov_2004, title={On the application of lacunae-based methods to Maxwell's equations}, volume={199}, DOI={10.1016/.jcp.2004.02.003}, number={1}, journal={Journal of Computational Physics}, author={Tsynkov, Semyon}, year={2004}, pages={126–149} }
 @article{tsynkov_2004, title={On the application of lacunae-based methods to Maxwell's equations}, volume={199}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2004.02.003}, DOI={10.1016/j.jcp.2004.02.003}, abstractNote={A straightforward application of the previously designed lacunae-based numerical methods to unsteady electromagnetic problems would encounter certain difficulties, as it may violate the continuity of the charges and currents, which is a necessary solvability condition for the Maxwell equations. In the paper, we prove existence of the special auxiliary charges and currents that satisfy the continuity equations identically. We also show that using such charges and currents as a part of the numerical procedure provides a clear and unobstructed venue toward implementation of the lacunae-based methods in electromagnetics.}, number={1}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Tsynkov, S.V.}, year={2004}, month={Sep}, pages={126–149} }
 @article{lončarić_tsynkov_2004, title={Optimization of power in the problems of active control of sound}, volume={65}, ISSN={0378-4754}, url={http://dx.doi.org/10.1016/j.matcom.2004.01.005}, DOI={10.1016/j.matcom.2004.01.005}, abstractNote={We analyze the problem of suppressing the unwanted component of a time-harmonic acoustic field (noise) on a predetermined region of interest. The suppression is rendered by active means, i.e., by introducing the additional acoustic sources called controls that generate the appropriate anti-sound. Previously, we have obtained general solutions for active controls in both continuous and discrete formulation of the problem. We have also obtained optimal solutions that minimize the L1 or L2 norm of the control sources; the physical interpretation of the former being the overall absolute acoustic source strength. In the current paper, we minimize the power required for the operation of the active control system. It turns out that the corresponding analysis necessarily involves interaction between the sources of sound and the surrounding acoustic field, which was not the case for either L1 or L2. Even though it may first seem counterintuitive, one can build a control system (a particular combination of surface monopoles and dipoles) that would require no power input for operation and would even produce a net power gain while providing the exact noise cancellation. This usually comes at the expense of having the original sources of noise produce even more energy.}, number={4-5}, journal={Mathematics and Computers in Simulation}, publisher={Elsevier BV}, author={Lončarić, J. and Tsynkov, S.V.}, year={2004}, month={May}, pages={323–335} }
 @article{tsynkov_2003, volume={18}, ISSN={0885-7474}, url={http://dx.doi.org/10.1023/A:1021111713715}, DOI={10.1023/A:1021111713715}, number={2}, journal={Journal of Scientific Computing}, publisher={Springer Nature}, author={Tsynkov, S. V.}, year={2003}, pages={155–189} }
 @book{tsynkov_2003, place={Raleigh, NC}, title={Artificial Boundary Conditions for the Numerical Simulation of
                   Unsteady Electromagnetic Waves}, url={https://stsynkov.math.ncsu.edu/publications/a34p.ncsu.pdf}, number={CRSC–TR03–19}, institution={Center for Research in Scientific Computation, North Carolina State University}, author={Tsynkov, S. V.}, year={2003} }
 @article{tsynkov_2003, title={Artificial boundary conditions for the numerical simulation of unsteady acoustic waves}, volume={189}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/S0021-9991(03)00249-3}, DOI={10.1016/S0021-9991(03)00249-3}, abstractNote={We construct non-local artificial boundary conditions (ABCs) for the numerical simulation of genuinely time-dependent acoustic waves that propagate from a compact source in an unbounded unobstructed space. The key property used for obtaining the ABCs is the presence of lacunae, i.e., sharp aft fronts of the waves, in wave-type solutions in odd-dimension spaces. This property can be considered a manifestation of the Huygens’ principle. The ABCs are obtained directly for the discrete formulation of the problem. They truncate the original unbounded domain and guarantee the complete transparency of the new outer boundary for all the outgoing waves. A central feature of the proposed ABCs is that the extent of their temporal non-locality is fixed and limited, and it does not come at the expense of simplifying the original model. It is rather a natural consequence of the existence of lacunae, which is a fundamental property of the corresponding solutions. The proposed ABCs can be built for any consistent and stable finite-difference scheme. Their accuracy can always be made as high as that of the interior approximation, and it will not deteriorate even when integrating over long time intervals. Besides, the ABCs are most flexible from the standpoint of geometry and can handle irregular boundaries on regular grids with no fitting/adaptation needed and no accuracy loss induced. Finally, they allow for a wide range of model settings. In particular, not only one can analyze the simplest advective acoustics case with the uniform background flow, but also the case when the waves’ source (or scatterer) is engaged in an accelerated motion.}, number={2}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Tsynkov, S.V.}, year={2003}, month={Aug}, pages={626–650} }
 @article{ilan_fibich_tsynkov_2003, title={Backscattering and Nonparaxiality Arrest Collapse of Damped Nonlinear Waves}, volume={63}, ISSN={0036-1399 1095-712X}, url={http://dx.doi.org/10.1137/S0036139902411855}, DOI={10.1137/S0036139902411855}, abstractNote={The critical nonlinear Schrodinger equation (NLS) models the propagation of intense laser light in Kerr media. This equation is derived from the more comprehensive nonlinear Helmholtz equation (NLH) by employing the paraxial approximation and neglecting the backscattered waves. It is known that if the input power of the laser beam (i.e., L2 norm of the initial solution) is sufficiently high, then the NLS model predicts that the beam will self-focus to a point (i.e., collapse) at a finite propagation distance. Mathematically, this behavior corresponds to the formation of a singularity in the solution of the NLS. A key question which has been open for many years is whether the solution to the NLH, i.e., the "parent" equation, may nonetheless exist and remain regular everywhere, particularly for those initial conditions (input powers) that lead to blowup in the NLS. In the current study we address this question by introducing linear damping into both models and subsequently comparing the numerical solutions of the damped NLH (boundary-value problem) with the corresponding solutions of the damped NLS (initial-value problem) for the case of one transverse dimension. Linear damping is introduced in much the same way as is done when analyzing the classical constant-coefficient Helmholtz equation using the limiting absorption principle. Numerically, we have found that it provides a very efficient tool for controlling the solutions of both the NLH and NLS. In particular, we have been able to identify initial conditions for which the NLS solution does become singular, while the NLH solution still remains regular everywhere. We believe that our finding of a larger domain of existence for the NLH than for the NLS is accounted for by precisely those mechanisms that have been neglected when deriving the NLS from the NLH, i.e., nonparaxiality and backscattering.}, number={5}, journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Ilan, B. and Fibich, G. and Tsynkov, S.}, year={2003}, month={Jan}, pages={1718–1736} }
 @inbook{tsynkov_2003, place={Berlin}, title={Lacunae-Based Artificial Boundary Conditions for the Numerical Simulation
             of Unsteady Waves Governed by Vector Models}, url={https://stsynkov.math.ncsu.edu/publications/waves2003p.pdf}, booktitle={Mathematical
             and Numerical Aspects of Wave Propagation --- WAVES 2003,
             The Sixth International Conference, Jyväskylä, Finland, June 30 -- July 4, 2003.
             Proceedings}, publisher={Springer}, author={Tsynkov, S. V.}, editor={Cohen, G. C. and Heikkola, E. and Joly, P. and Neittaanmäki, P.Editors}, year={2003}, pages={103–108} }
 @inbook{lončarić_tsynkov_2003, title={Optimization in the Context of Active Control of Sound}, volume={2668}, ISBN={9783540401612 9783540448433}, ISSN={0302-9743}, url={http://dx.doi.org/10.1007/3-540-44843-8_87}, DOI={10.1007/3-540-44843-8_87}, abstractNote={A problem of eliminating the unwanted time-harmonic noise on a predetermined region of interest is solved by active means, i.e., by introducing the additional sources of sound, called controls, that generate the appropriate annihilating signal (anti-sound). The general solution for controls has been obtained previously for both the continuous and discrete formulation of the problem. Next, the control sources are optimized using different criteria. Minimization of the overall absolute acoustic source strength is equivalent to minimization of multi-variable complex functions in the sense of L 1 with conical constraints. The global L 1 optimum appears to be a special layer of monopoles on the perimeter of the protected region. The use of quadratic cost functions, e.g., the L 2 norm of the controls, leads to a versatile numerical procedure. It allows one to analyze sophisticated geometries in the case of a constrained minimization. Finally, minimization of power consumed by an active control system always involves interaction between the sources of sound and the surrounding acoustic field, which was not the case for either L 1 or L 2. One can, in fact, build a control system that would require no power input for operation and may even produce a net power gain while providing the exact noise cancellation. This, of course, comes at the expense of having the original sources of noise produce even more energy.}, booktitle={Computational Science and Its Applications — ICCSA 2003}, publisher={Springer Berlin Heidelberg}, author={Lončarić, Josip and Tsynkov, Semyon}, year={2003}, pages={801–810} }
 @article{loncaric_tsynkov_2003, title={Optimization of Acoustic Source Strength in the Problems of Active Noise Control}, volume={63}, ISSN={0036-1399 1095-712X}, url={http://dx.doi.org/10.1137/S0036139902404220}, DOI={10.1137/S0036139902404220}, abstractNote={We consider a problem of eliminating the unwanted time-harmonic noise on a pre- determined region of interest. The desired objective is achieved by active means, i.e., by introducing additional sources of sound called control sources, which generate the appropriate annihilating acous- tic signal (antisound). A general solution for the control sources has been obtained previously in both continuous and discrete formulation of the problem. In the current paper, we focus on optimizing the overall absolute acoustic source strength of the control sources. Mathematically, this amounts to the minimization of multivariable complex-valued functions in the sense of L1 with conical constraints, which are only "marginally" convex. The corresponding numerical optimization problem appears very challenging even for the most sophisticated state-of-the-art methodologies, and even when the dimension of the grid is small and the waves are long. Our central result is that the global L1-optimal solution can, in fact, be obtained without solving the numerical optimization problem. This solution is given by a special layer of monopole sources on the perimeter of the protected region. We provide a rigorous proof of global L1 minimality for both continuous and discrete optimization problems in the one-dimensional case. We also provide numerical evidence that corroborates our result in the two-dimensional case, when the protected domain is a cylinder. Even though we cannot fully justify it, we believe that the same result holds in the general case, i.e., for multidimensional settings and domains of arbitrary shape. We formulate this notion as a conjecture at the end of the paper.}, number={4}, journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Loncaric, J. and Tsynkov, S. V.}, year={2003}, month={Jan}, pages={1141–1183} }
 @book{abarbanel_tsynkov_turkel_2002, place={Hampton, VA}, title={A Future Role of Numerical and Applied Mathematics in Material Sciences}, url={https://apps.dtic.mil/dtic/tr/fulltext/u2/a402142.pdf}, number={40, NASA/CR–2002–211453}, institution={ICASE}, author={Abarbanel, S. and Tsynkov, S. and Turkel, E.}, year={2002} }
 @article{fibich_ilan_tsynkov_2002, title={Computation of Nonlinear Backscattering Using a High-Order Numerical Method}, volume={17}, ISSN={0885-7474}, url={http://dx.doi.org/10.1023/a:1015181404953}, DOI={10.1023/a:1015181404953}, number={1-4}, journal={Journal of Scientific Computing}, publisher={Springer Nature}, author={Fibich, G. and Ilan, B. and Tsynkov, S.}, year={2002}, pages={351–364} }
 @inbook{ryaben’kii_2002, place={Berlin}, title={On the Results of the Application of the Method of Difference Potentials to the Construction of Artificial Boundary Conditions for External Flow Computations}, volume={30}, ISBN={9783642627156 9783642563447}, ISSN={0179-3632}, url={http://dx.doi.org/10.1007/978-3-642-56344-7_17}, DOI={10.1007/978-3-642-56344-7_17}, abstractNote={Let us first briefly repeat the general arguments behind constructing the artificial boundary conditions (ABCs) for the numerical solution of problems formulated on unbounded domains. As has been mentioned, a standard approach to solving infinite-domain boundary-value problems on the computer involves truncation as a first step, prior to the discretization of the continuous problem and solution of the resulting discrete system. The truncated problem is clearly subdefinite unless supplemented by the proper closing procedure at the external boundary of the finite computational domain. The latter boundary is often called artificial emphasizing the fact that it originates from the numerical limitations rather than original physical formulation. The corresponding closing procedure is called the ABCs.}, booktitle={Method of Difference Potentials and Its Applications}, publisher={Springer Berlin Heidelberg}, author={Ryaben’kii, Viktor S.}, year={2002}, pages={403–441} }
 @article{roberts_sidilkover_tsynkov_2002, title={On the combined performance of nonlocal artificial boundary conditions with the new
         generation of advanced multigrid flow solvers}, volume={31}, url={http://www.sciencedirect.com/science/article/pii/S0045793001000457}, DOI={https://doi.org/10.1016/S0045-7930(01)00045-7}, abstractNote={We develop theoretically and implement numerically a unified flow solution methodology that combines the advantages relevant to two independent groups of methods in computational fluid dynamics that have recently proven successful: The new factorizable schemes for the equations of hydrodynamics that facilitate the construction of optimally convergent multigrid algorithms, and highly accurate global far-field artificial boundary conditions (ABCs). The primary result that we have obtained is the following. Global ABCs do not hamper the optimal (i.e., unimprovable) multigrid convergence rate pertinent to the solver. At the same time, contrary to the standard local ABCs, the solution accuracy provided by the global ABCs deteriorates very slightly or does not deteriorate at all when the computational domain shrinks, which clearly translates into substantial savings of computer resources.}, number={3}, journal={Computers & Fluids}, author={Roberts, T.W. and Sidilkover, D. and Tsynkov, S.V.}, year={2002}, pages={269–308} }
 @article{roberts_sidilkover_tsynkov_2002, title={On the combined performance of nonlocal artificial boundary conditions with the new generation of advanced multigrid flow solvers}, volume={31}, ISSN={0045-7930}, url={http://dx.doi.org/10.1016/S0045-7930(01)00045-7}, DOI={10.1016/S0045-7930(01)00045-7}, abstractNote={We develop theoretically and implement numerically a unified flow solution methodology that combines the advantages relevant to two independent groups of methods in computational fluid dynamics that have recently proven successful: The new factorizable schemes for the equations of hydrodynamics that facilitate the construction of optimally convergent multigrid algorithms, and highly accurate global far-field artificial boundary conditions (ABCs). The primary result that we have obtained is the following. Global ABCs do not hamper the optimal (i.e., unimprovable) multigrid convergence rate pertinent to the solver. At the same time, contrary to the standard local ABCs, the solution accuracy provided by the global ABCs deteriorates very slightly or does not deteriorate at all when the computational domain shrinks, which clearly translates into substantial savings of computer resources.}, number={3}, journal={Computers & Fluids}, publisher={Elsevier BV}, author={Roberts, T.W. and Sidilkover, D. and Tsynkov, S.V.}, year={2002}, month={Mar}, pages={269–308} }
 @inbook{tsynkov_turkel_2001, place={Huntington, NY}, title={A Cartesian Perfectly Matched Layer for the
                   ̆ppercaseHelmholtz Equation}, url={https://stsynkov.math.ncsu.edu/publications/a24e7.pdf}, booktitle={Absorbing Boundaries and Layers, Domain Decomposition Methods.
                   ̆ppercaseApplications to Large Scale Computations}, publisher={Nova Science Publishers, Inc.}, author={Tsynkov, S. V. and Turkel, E.}, editor={Tourrette Loı̈c and Halpern, LoranceEditors}, year={2001}, pages={279–309} }
 @inbook{tsynkov_turkel_2001, place={New York}, title={A Cartesian Perfectly Matched Layer for the Helmholtz Equation}, booktitle={Absorbing Boundaries and Layers, Domain Decomposition Methods. Applications to Large Scale Computations}, publisher={Nova Science Publishers}, author={Tsynkov, S.V. and Turkel, E.}, editor={Tourrette Loı̈c and Halpern, LoranceEditors}, year={2001}, pages={279–309} }
 @article{loncaric_ryaben'kii_tsynkov_2001, title={Active Shielding and Control of Noise}, volume={62}, ISSN={0036-1399 1095-712X}, url={http://dx.doi.org/10.1137/s0036139900367589}, DOI={10.1137/s0036139900367589}, abstractNote={We present a mathematical framework for the active control of time-harmonic acoustic disturbances. Unlike many existing methodologies, our approach provides for the exact volumetric cancellation of unwanted noise in a given predetermined region of space while leaving unaltered those components of the total acoustic field that are deemed friendly. Our key finding is that for eliminating the unwanted component of the acoustic field in a given area, one needs to know relatively little; in particular, neither the locations nor structure nor strength of the exterior noise sources need to be known. Likewise, there is no need to know the volumetric properties of the supporting medium across which the acoustic signals propagate, except, perhaps, in the narrow area of space near the boundary (perimeter) of the domain to be shielded. The controls are built based solely on the measurements performed on the perimeter of the region to be shielded; moreover, the controls themselves (i.e., additional sources) are also c...}, number={2}, journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Loncaric, J. and Ryaben'kii, V. S. and Tsynkov, S. V.}, year={2001}, month={Jan}, pages={563–596} }
 @article{ryaben'kii_tsynkov_turchaninov_2001, title={Global Discrete Artificial Boundary Conditions for Time-Dependent Wave Propagation}, volume={174}, ISSN={0021-9991}, url={http://dx.doi.org/10.1006/jcph.2001.6936}, DOI={10.1006/jcph.2001.6936}, abstractNote={We construct global artificial boundary conditions (ABCs) for the numerical simulation of wave processes on unbounded domains using a special non-deteriorating algorithm that has been developed previously for the long-term computation of wave-radiation solutions. The ABCs are obtained directly for the discrete formulation of the problem; in so doing, neither a rational approximation of ``non-reflecting kernels," nor discretization of the continuous boundary conditions is required. The extent of temporal nonlocality of the new ABCs appears fixed and limited; in addition, the ABCs can handle artificial boundaries of irregular shape on regular grids with no fitting/adaptation needed and no accuracy loss induced. The non-deteriorating algorithm, which is the core of the new ABCs, is inherently three-dimensional, it guarantees temporally uniform grid convergence of the solution driven by a continuously operating source on arbitrarily long time intervals, and provides unimprovable linear computational complexity with respect to the grid dimension. The algorithm is based on the presence of lacunae, i.e., aft fronts of the waves, in wave-type solutions in odd-dimension spaces. It can, in fact, be built as a modification on top of any consistent and stable finite-difference scheme, making its grid convergence uniform in time and at the same time keeping the rate of convergence the same as that of the non-modified scheme. In the paper, we delineate the construction of the global lacunae-based ABCs in the framework of a discretized wave equation. The ABCs are obtained for the most general formulation of the problem that involves radiation of waves by moving sources (e.g., radiation of acoustic waves by a maneuvering aircraft). We also present systematic numerical results that corroborate the theoretical design properties of the ABCs'' algorithm.}, number={2}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Ryaben'kii, V.S. and Tsynkov, S.V. and Turchaninov, V.I.}, year={2001}, month={Dec}, pages={712–758} }
 @article{fibich_tsynkov_2001, title={High-Order Two-Way Artificial Boundary Conditions for Nonlinear Wave Propagation with Backscattering}, volume={171}, ISSN={0021-9991}, url={http://dx.doi.org/10.1006/jcph.2001.6800}, DOI={10.1006/jcph.2001.6800}, abstractNote={Abstract. When solving linear scattering problems, one typically first solves for the impinging wavein the absence of obstacles. Then, using the linear superposition principle, the original problem is reducedto one which involves only the scattered wave (which is driven by the values of the impinging field at thesurface of the obstacles). When the original domain is unbounded, special artificial boundary conditions(ABCs) have to be set at the outer (artificial) boundary of the finite computational domain, in order toguarantee the reflectionless propagation of waves through this external artificial boundary. The situationbecomes conceptually different when the propagation equation is nonlinear. In this case the impinging andscattered waves can no longer be separated, and the problem has to be solved in its entirety. In particular, theboundary on which the incoming field values are prescribed, should transmit the given incoming waves in onedirection and simultaneously be transparent to all the outgoing waves that travel in the opposite direction.We call this type of boundary conditions two-way ABCs. In the paper, we construct the two-way ABCsfor the nonlinear Helmholtz equation, which models a continuous-wave (CW) laser beam propagation in amedium with nonlinear index of refraction. In this case, the forward propagation of the beam is accompaniedby backscattering, i.e., generation of waves in the opposite direction to that of the incoming signal. Our two-way ABCs generate no reflection of the backscattered waves and at the same time impose the correct valuesof the incoming wave. The ABCs are obtained in the framework of a fourth-order accurate discretization tothe Helmholtz operator inside the computational domain. The fourth-order convergence of our methodologyis corroborated experimentally by solving linear model problems. We also present solutions in the nonlinearcase using the two-way ABC which, unlike the traditional Dirichlet boundary condition approach, allows fordirect calculation of the magnitude of backscattering.Key words, artificial boundary conditions (ABCs), two-way ABCs, radiation, the Helmholtz equation,nonlinearity, nonparaxiality, fourth-order schemes, self-focusing, backscatteringSubject classification. Applied and Numerical Mathematics1. Introduction. The Helmholtz equation02 0 2AE(Xl,...,XD) + k2E = O, A = _ +... + Ox_ ' (1.1)}, number={2}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Fibich, Gadi and Tsynkov, Semyon}, year={2001}, month={Aug}, pages={632–677} }
 @article{ryaben'kii_tsynkov_turchaninov_2001, title={Long-time numerical computation of wave-type solutions driven by moving sources}, volume={38}, ISSN={0168-9274}, url={http://dx.doi.org/10.1016/S0168-9274(01)00038-1}, DOI={10.1016/S0168-9274(01)00038-1}, abstractNote={Abstract We propose a methodology for calculating the solution of an initial-value problem for the three-dimensional wave equation over arbitrarily long time intervals. The solution is driven by moving sources that are compactly supported in space for any particular moment of time; the extent of the support is assumed bounded for all times. By a simple change of variables the aforementioned formulation obviously translates into the problem of propagation of waves across a medium in motion, which has multiple applications in unsteady aerodynamics, advective acoustics, and other areas. The algorithm constructed in the paper can employ any appropriate (i.e., consistent and stable) explicit finite-difference scheme for the wave equation. This scheme is used as a core computational technique and modified so that to allow for a non-deteriorating calculation of the solution for as long as necessary. Provided that the original underlying scheme converges in some sense, i.e., in suitable norms with a particular rate, we prove the grid convergence of the new algorithm in the same sense uniformly in time on arbitrarily long intervals. Thus, the new algorithm obviously does not accumulate error in the course of time; besides, it requires only a fixed non-growing amount of computer resources (memory and processor time) per one time step; these amounts are linear with respect to the grid dimension and thus optimal. The algorithm is inherently three-dimensional; it relies on the presence of lacunae in the solutions of the wave equation in odd-dimension spaces. The methodology presented in the paper is, in fact, a building block for constructing the nonlocal highly accurate unsteady artificial boundary conditions to be used for the numerical simulation of waves propagating with finite speed over unbounded domains.}, number={1-2}, journal={Applied Numerical Mathematics}, publisher={Elsevier BV}, author={Ryaben'kii, V.S. and Tsynkov, S.V. and Turchaninov, V.I.}, year={2001}, month={Jul}, pages={187–222} }
 @article{tsynkov_abarbanel_nordström_ryaben'kii_vatsa_2000, title={Global Artificial Boundary Conditions for Computation of External Flows with Jets}, volume={38}, ISSN={0001-1452 1533-385X}, url={http://dx.doi.org/10.2514/2.888}, DOI={10.2514/2.888}, abstractNote={We propose new global artificial boundary conditions (ABCs) for computation of flows with propulsive jets. The algorithm is based on application of the difference potentials method (DPM). Previously, similar boundary conditions have been implemented for calculation of external compressible viscous flows around finite bodies. The proposed modification substantially extends the applicability range of the DPM-based algorithm. We present the general formulation of the problem, describe our numerical methodology, and discuss the corresponding computational results. The particular configuration that we analyze is a slender three-dimensional body with boat-tail geometry and supersonic jet exhaust in a subsonic external flow under zero angle of attack. Similar to the results obtained earlier for the flows around airfoils and wings, current results for the jet flow case corroborate the superiority of the DPM-based ABCs over standard local methodologies from the standpoints of accuracy, overall numerical performance, and robustness}, number={11}, journal={AIAA Journal}, publisher={American Institute of Aeronautics and Astronautics (AIAA)}, author={Tsynkov, Semyon and Abarbanel, Saul and Nordström, Jan and Ryaben'kii, Victor and Vatsa, Veer}, year={2000}, month={Nov}, pages={2014–2022} }
 @article{tsynkov_abarbanel_nordstrom_ryaben'kii_vasta_2000, title={Global artificial boundary conditions for computation of external flows with jets}, volume={38}, ISSN={0001-1452 1533-385X}, url={http://dx.doi.org/10.2514/3.14645}, DOI={10.2514/3.14645}, journal={AIAA Journal}, publisher={American Institute of Aeronautics and Astronautics (AIAA)}, author={Tsynkov, Semyon and Abarbanel, Saul and Nordstrom, Jan and Ryaben'kii, Victor and Vasta, Veer}, year={2000}, month={Jan}, pages={2014–2022} }
 @article{ryaben’kii_turchaninov_tsynkov_2000, title={Non-Reflecting Artificial Boundary Conditions for the Replacement of Truncated Equations with Lacunae}, volume={12}, number={12}, journal={Mathematical Modeling}, author={Ryaben’kii, V.S. and Turchaninov, V.I. and Tsynkov, S.V.}, year={2000}, pages={108–127} }
 @article{nonreflecting artificial boundary conditions for the
              replacement of rejected equations with gaps_2000, volume={12}, url={https://mathscinet.ams.org/mathscinet-getitem?mr=1833776}, note={[in Russian]}, number={12}, journal={Mat. Model.}, year={2000}, pages={108–127} }
 @article{tsynkov_1999, title={External Boundary Conditions for Three-Dimensional Problems of Computational Aerodynamics}, volume={21}, ISSN={1064-8275 1095-7197}, url={http://dx.doi.org/10.1137/s1064827597318757}, DOI={10.1137/s1064827597318757}, abstractNote={We consider an unbounded steady-state flow of viscous fluid over a three-dimensional finite body or configuration of bodies. For the purpose of solving this flow numerically, we discretize the governing equations (Navier--Stokes) on a finite-difference grid. Prior to the discretization, we obviously need to truncate the original unbounded domain by introducing an artificial computational boundary at a finite distance from the body; otherwise, the number of discrete variables will not be finite. This artificial boundary is typically the external boundary of the domain covered by the grid. 
The flow problem (both continuous and discretized) formulated on the finite computational domain is clearly subdefinite unless supplemented by some artificial boundary conditions (ABCs) at the external computational boundary. In this paper, we present an innovative approach to constructing highly accurate ABCs for three-dimensional flow computations. The approach extends our previous technique developed for the two-dimensional case; it employs the finite-difference counterparts to Calderon's pseudodifferential boundary projections calculated in the framework of the difference potentials method (DPM) of Ryaben'kii. The resulting ABCs appear spatially nonlocal but are particularly easy to implement along with the existing flow solvers. 
The new boundary conditions have been successfully combined with the NASA-developed production code TLNS3D and used for the analysis of wing-shaped configurations in subsonic and transonic flow regimes. As demonstrated by the computational experiments and comparison with the standard local methods, the DPM-based ABCs allow one to greatly reduce the size of the computational domain while still maintaining high accuracy of the numerical solution. Moreover, they may provide for a noticeable speedup of multigrid convergence.}, number={1}, journal={SIAM Journal on Scientific Computing}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Tsynkov, Semyon V.}, year={1999}, month={Jan}, pages={166–206} }
 @inproceedings{tsynkov_abarbanel_nordstrom_ryaben'kii_vatsa_1999, title={Global artificial boundary conditions for computation of external flow problems with propulsive jets}, volume={2}, url={http://dx.doi.org/10.2514/6.1999-3351}, DOI={10.2514/6.1999-3351}, abstractNote={We propose new global artificial boundary conditions (ABC''s) for computation of flows with propulsive jets. The algorithm is based on application of the difference potentials method (DPM). Previously, similar boundary conditions have been implemented for calculation of external compressible viscous flows around finite bodies. The proposed modification substantially extends the applicability range of the DPM-based algorithm. In the paper, we present the general formulation of the problem, describe our numerical methodology, and discuss the corresponding computational results. The particular configuration that we analyze is a slender three-dimensional body with boat-tail geometry and supersonic jet exhaust in a subsonic external flow under zero angle of attack. Similarly to the results obtained earlier for the flows around airfoils and wings, current results for the jet flow case corroborate the superiority of the DPM-based ABC''s over standard local methodologies from the standpoints of accuracy, overall numerical performance, and robustness.}, note={AIAA Paper No. 99–3351}, booktitle={14th Computational Fluid Dynamics Conference}, publisher={American Institute of Aeronautics and Astronautics}, author={Tsynkov, Semyon and Abarbanel, Saul and Nordstrom, Jan and Ryaben'kii, Victor and Vatsa, Veer}, year={1999}, month={Nov}, pages={836–846} }
 @book{ryaben’kii_turchaninov_tsynkov_1999, place={Hampton, VA}, title={Long-Time Numerical Integration of the Three-Dimensional Wave
                    Equation in the Vicinity of a Moving Source}, url={https://ntrs.nasa.gov/search.jsp?R=19990062251}, number={99–23, NASA/CR–1999–209350}, institution={ICASE}, author={Ryaben’kii, V. S. and Turchaninov, V. I. and Tsynkov, S. V.}, year={1999} }
 @article{ryaben’kii_turchaninov_tsynkov_1999, title={On Lacunae-Based Algorithm for Numerical Solution of 3D Wave Equation for Arbitrarily Large Time}, volume={11}, number={12}, journal={Mathematical Modeling}, author={Ryaben’kii, V.S. and Turchaninov, V.I. and Tsynkov, S.V.}, year={1999}, pages={113–127} }
 @article{the use of lacunae of the 3d-wave equation for computing a
              solution at large time values_1999, volume={11}, url={https://mathscinet.ams.org/mathscinet-getitem?mr=1761052}, note={[in Russian]}, number={12}, journal={Mat. Model.}, year={1999}, pages={113–126} }
 @inbook{ryaben'kii_tsynkov_1998, place={Singapore}, title={AN APPLICATION OF THE DIFFERENCE POTENTIALS METHOD TO SOLVING EXTERNAL PROBLEMS IN CFD}, volume={1}, ISBN={9789810235642 9789812812957}, url={http://dx.doi.org/10.1142/9789812812957_0010}, DOI={10.1142/9789812812957_0010}, abstractNote={Numerical solution of infinite-domain boundary-value problems requires some special techniques that would make the problem available for treatment on the computer. Indeed, the problem must be discretized in a way that the computer operates with only finite amount of information. Therefore, the original infinite-domain formulation must be altered and/or augmented so that on one hand the solution is not changed (or changed slightly) and on the other hand the finite discrete formulation becomes available. One widely used approach to constructing such discretizations consists of truncating the unbounded original domain and then setting the artificial boundary conditions (ABC''s) at the newly formed external boundary. The role of the ABC''s is to close the truncated problem and at the same time to ensure that the solution found inside the finite computational domain would be maximally close to (in the ideal case, exactly the same as) the corresponding fragment of the original infinite-domain solution. Let us emphasize that the proper treatment of artificial boundaries may have a profound impact on the overall quality and performance of numerical algorithms. The latter statement is corroborated by the numerous computational experiments and especially concerns the area of CFD, in which external problems present a wide class of practically important formulations. In this paper, we review some work that has been done over the recent years on constructing highly accurate nonlocal ABC''s for calculation of compressible external flows. The approach is based on implementation of the generalized potentials and pseudodifferential boundary projection operators analogous to those proposed first by Calderon. The difference potentials method (DPM) by Ryaben''kii is used for the effective computation of the generalized potentials and projections. The resulting ABC''s clearly outperform the existing methods from the standpoints of accuracy and robustness, in many cases noticeably speed up the multigrid convergence, and at the same time are quite comparable to other methods from the standpoints of geometric universality and easiness of implementation.}, booktitle={Computational Fluid Dynamics Review 1998}, publisher={WORLD SCIENTIFIC}, author={Ryaben'kii, Victor S. and Tsynkov, Semyon V.}, editor={Hafez, M. and Oshima, K.Editors}, year={1998}, month={Nov}, pages={169–205} }
 @inbook{tsynkov_1998, title={Artificial Boundary Conditions for Infinite-Domain Problems}, volume={6}, ISBN={9789401061735 9789401151696}, ISSN={1381-1339}, url={http://dx.doi.org/10.1007/978-94-011-5169-6_7}, DOI={10.1007/978-94-011-5169-6_7}, abstractNote={We present a new approach to constricting artificial boundary conditions for calculating three-dimensional external flows over finite bodies. The approach is based on application of the difference potentials method by V. S. Ryaben’kii and extends our previous technique developed for the two-dimensional case.}, booktitle={ICASE/LaRC Interdisciplinary Series in Science and Engineering}, publisher={Springer Netherlands}, author={Tsynkov, Semyon V.}, year={1998}, pages={119–137} }
 @article{tsynkov_vatsa_1998, title={Improved Treatment of External Boundary Conditions for Three-Dimensional Flow Computations}, volume={36}, ISSN={0001-1452 1533-385X}, url={http://dx.doi.org/10.2514/2.327}, DOI={10.2514/2.327}, abstractNote={An innovative numerical approach is presented for setting highly accurate nonlocal boundary conditions at the external computational boundaries when calculating three-dimensional compressible viscous flows over finite bodies. The approach is based on application of the special boundary operators analogous to Calderon's projections (Calderon, A. P.) and the difference potentials method by Ryaben'kii; it extends the previous technique developed for the two-dimensional case. The new boundary conditions methodology has been successfully combined with the NASA-developed code TLNS3D and used for the analysis of wing-shaped configurations in subsonic and transonic flow regimes. As demonstrated by the computational experiments, the improved external boundary conditions allow one to greatly reduce the size of the computational domain while still maintaining high accuracy of the numerical solution. Moreover, they may provide for a noticeable speedup of convergence of the multigrid iterations}, number={11}, journal={AIAA Journal}, publisher={American Institute of Aeronautics and Astronautics (AIAA)}, author={Tsynkov, Semyon V. and Vatsa, Veer N.}, year={1998}, month={Nov}, pages={1998–2004} }
 @article{tsynkov_1998, title={Numerical solution of problems on unbounded domains. A review}, volume={27}, ISSN={0168-9274}, url={http://dx.doi.org/10.1016/s0168-9274(98)00025-7}, DOI={10.1016/s0168-9274(98)00025-7}, abstractNote={While numerically solving a problem initially formulated on an unbounded domain, one typically truncates this domain, which necessitates setting the artificial boundary conditions (ABCs) at the newly formed external boundary. The issue of setting the ABCs appears most significant in many areas of scientific computing, for example, in problems originating from acoustics, electrodynamics, solid mechanics, and fluid dynamics. In particular, in computational fluid dynamics (where external problems represent a wide class of important formulations) the proper treatment of external boundaries may have a profound impact on the overall quality and performance of numerical algorithms and interpretation of the results. Most of the currently used techniques for setting the ABCs can basically be classified into two groups. The methods from the first group (global ABCs) usually provide high accuracy and robustness of the numerical procedure but often appear to be fairly cumbersome and (computationally) expensive. The methods from the second group (local ABCs) are, as a rule, algorithmically simple, numerically cheap, and geometrically universal; however, they usually lack accuracy of computations. In this paper we first present an extensive survey and provide a comparative assessment of different existing methods for constructing the ABCs. Then, we describe a new ABCs technique proposed in our recent work and review the corresponding results. This new technique enables one to construct the ABCs that largely combine the advantages relevant to the two aforementioned classes of existing methods. Our approach is based on application of the difference potentials method by Ryaben'kii. This approach allows one to obtain highly accurate ABCs in the form of certain (nonlocal) boundary operator equations. The operators involved are analogous to the pseudodifferential boundary projections first introduced by Calderon and then also studied by Seeley. In spite of the nonlocality, the new boundary conditions are geometrically universal, numerically inexpensive, and easy to implement along with the existing solvers.}, note={Absorbing boundary conditions}, number={4}, journal={Applied Numerical Mathematics}, publisher={Elsevier BV}, author={Tsynkov, Semyon V.}, year={1998}, month={Aug}, pages={465–532} }
 @inbook{tsynkov_1998, place={Dordrecht}, title={On the Combined Implementation of Global Boundary Conditions with Central Difference Multigrid Flow Solvers}, volume={49}, ISBN={9789048151066 9789401590952}, ISSN={0926-5112}, url={http://dx.doi.org/10.1007/978-94-015-9095-2_31}, DOI={10.1007/978-94-015-9095-2_31}, abstractNote={In modern scientific computing, multigrid methods have proven to provide for one of the most efficient ways for the iterative solution of large algebraic systems that arise from discretization of the original continuous problems. These methods have been widely and successfully implemented over the last fifteen years in different applied areas, in particular, computational fluid dynamics (CFD). The key idea behind the multigrid methodology (that the short-wave components of the error expansion on the grid decay faster than the long-wave ones and therefore, the problem should be considered on a sequence of grids so that for any wave there be a grid, on which this wave can be regarded as short) appears quite universal, although every particular case may require working out some specific issues relevant to the particular type of implementation considered.}, booktitle={Fluid Mechanics and Its Applications}, publisher={Springer Netherlands}, author={Tsynkov, Semyon V.}, editor={Geers, Thomas L.Editor}, year={1998}, pages={285–294} }
 @article{tsynkov_1997, title={Artificial Boundary Conditions for Computation of Oscillating External Flows}, volume={18}, ISSN={1064-8275 1095-7197}, url={http://dx.doi.org/10.1137/s1064827595291145}, DOI={10.1137/s1064827595291145}, abstractNote={In this paper, we propose a new technique for the numerical treatment of external flow problems with oscillatory behavior of the solution in time. Specifically, we consider the case of unbounded compressible viscous plane flow past a finite body (airfoil). Oscillations of the flow in time may be caused, for example, by the time-periodic injection of fluid into the boundary layer, which in accordance with experimental data, may essentially increase the performance of the airfoil. 
To conduct the actual computations, we have to somehow restrict the original unbounded domain, that is, to introduce an artificial (external) boundary and to further consider only a finite computational domain. Consequently, we will need to formulate some artificial boundary conditions (ABCs) at the introduced external boundary. The ABCs we are aiming to obtain must meet the following fundamental requirement. One should be able to uniquely complement the solution calculated inside the finite computational domain to its infinite exterior so that the original problem is solved within the desired accuracy. 
Our construction of such ABCs for oscillating flows is based on an essential assumption: the Navier--Stokes equations can be linearized in the far field against the free-stream background. To actually compute the ABCs, we represent the far-field solution as a Fourier series in time and then apply the difference potentials method (DPM) of V. S. Ryaben'kii. 
This paper contains a general theoretical description of the algorithm for setting the DPM-based ABCs for time-periodic external flows. Based on our experience in implementing analogous ABCs for steady-state problems (a simpler case), we expect that these boundary conditions will become an effective tool for constructing robust numerical methods to calculate oscillatory flows.}, number={6}, journal={SIAM Journal on Scientific Computing}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Tsynkov, S. V.}, year={1997}, month={Nov}, pages={1612–1656} }
 @book{tsynkov_1996, place={Hampton, VA}, title={Artificial Boundary Conditions Based on the Difference Potentials Method}, url={https://ntrs.nasa.gov/search.jsp?R=19960045440}, number={NASA-TM-110265, NAS 1.15:110265}, institution={NASA Langley Research Center}, author={Tsynkov, S. V.}, year={1996} }
 @article{tsynkov_1996, title={Construction of Artificial Boundary Conditions Using Difference Potentials Method}, volume={8}, url={https://mathscinet.ams.org/mathscinet-getitem?mr=1444877}, number={9}, journal={Mathematical Modeling}, author={Tsynkov, S.V.}, year={1996}, pages={118–128} }
 @article{tsynkov_turkel_abarbanel_1996, title={External flow computations using global boundary conditions}, volume={34}, ISSN={0001-1452 1533-385X}, url={http://dx.doi.org/10.2514/3.13130}, DOI={10.2514/3.13130}, abstractNote={We numerically integrate the compressible Navier-Stokes equations by means of a finite volume technique on the domain exterior to an airfoil. The curvilinear grid we use for discretization of the Navier-Stokes equations is obviously finite, it covers only a certain bounded region around the airfoil, consequently, we need to set some artificial boundary conditions at the external boundary of this region. The artificial boundary conditions we use here are non-local in space. They are constructed specifically for the case of steady-state solution. In constructing the artificial boundary conditions, we linearize the Navier-Stokes equations around the far-field solution and apply the difference potentials method. The resulting global conditions are implemented together with a pseudotime multigrid iteration procedure for achieving the steady state. The main goal of this paper is to describe the numerical procedure itself, therefore, we primarily emphasize the computation of artificial boundary conditions and the combined usage of these artificial boundary conditions and the original algorithm for integrating the Navier-Stokes equations. The underlying theory that justifies the proposed numerical techniques will accordingly be addressed more briefly.}, number={4}, journal={AIAA Journal}, publisher={American Institute of Aeronautics and Astronautics (AIAA)}, author={Tsynkov, S. V. and Turkel, E. and Abarbanel, S.}, year={1996}, month={Apr}, pages={700–706} }
 @inproceedings{tsynkov_1996, place={Chichester}, title={Nonlocal Artificial Boundary Conditions
               for Computation of External Viscous Flows}, url={https://www.tib.eu/en/search/id/BLCP%3ACN017470620/Nonlocal-Artificial-Boundary-Conditions-for-Computation/}, booktitle={Computational Fluid Dynamics'96.
                Proceedings of the Third ECCOMAS CFD Conference,
                September 9--13, 1996, Paris, France}, publisher={John Wiley & Sons}, author={Tsynkov, S. V.}, editor={Desideri, J.-A. and Hirsch, C. and Le Tallec, P. and Pandolfi, M. and Périaux, J.Editors}, year={1996}, pages={512–518} }
 @article{tsynkov_1995, title={An Application of Nonlocal External Conditions to Viscous Flow Computations}, volume={116}, ISSN={0021-9991}, url={http://dx.doi.org/10.1006/jcph.1995.1022}, DOI={10.1006/jcph.1995.1022}, abstractNote={We are looking for a steady-state solution of an external flow problem originally formulated on an unbounded domain. Our case is a 2D viscous compressible flow past a finite body (airfoil). We truncate the original domain by introducing a finite grid around the airfoil and integrate the Navier-Stokes equations on this grid with the help of a finite-volume code which involves a multigrid pseudo-time iteration technique for achieving a steady state. To integrate the Navier?Stokes equations on a finite subregion of an original domain only we supplement the numerical algorithm by special nonlocal artificial boundary conditions formulated on an external boundary of the finite computational domain. These artificial boundary conditions are based on the difference potentials method proposed by V. S. Ryaben'kii. We compare the results provided by the nonlocal conditions with those obtained from the standard external conditions which are based on locally one-dimensional characteristic analysis at inflow and extrapolation at outflow. It turns out that the nonlocal artificial boundary conditions accelerate the convergence by about a factor of 3, as well as allow one to shrink substantially the computational domain without loss of accuracy.}, number={2}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Tsynkov, S.V.}, year={1995}, month={Feb}, pages={212–225} }
 @article{ryaben'kii_tsynkov_1995, title={An effective numerical technique for solving a special class of ordinary difference equations}, volume={18}, ISSN={0168-9274}, url={http://dx.doi.org/10.1016/0168-9274(95)00081-5}, DOI={10.1016/0168-9274(95)00081-5}, abstractNote={We consider a system of ordinary difference equations with constant coefficients, which is defined on an infinite one-dimensional mesh. The right-hand side (RHS) of the system is compactly supported, therefore, the system appears to be homogeneous outside some finite mesh interval. At infinity, we impose certain boundary conditions, e.g., conditions of boundedness or decay of the solution, so that the resulting boundary-value problem is uniquely solvable and well posed. We also consider a truncation of this infinite-domain problem to some finite mesh interval that entirely contains the support of the RHS. We require that the solution to this truncated problem, which is the one we are going to actually calculate, coincides on the finite mesh interval where it is defined with the corresponding fragment of the solution to the original (infinite) problem. This requirement necessitates setting some special boundary conditions at the ends of the aforementioned finite interval. In so doing, one should guarantee an exact transfer of boundary conditions from infinity through the (semi-infinite) intervals of homogeneity of the original system. It turns out that the desired boundary conditions at the ends of the finite interval can be naturally formulated in terms of the eigen subspaces of the system operator. This, in turn, enables us to develop an effective numerical algorithm for solving the system of ordinary difference equations on the finite mesh interval. This algorithm can be referred to as a version of the well-known successive substitution technique but without its final (“inverse” or “resolving”) stage. The special class of systems described in this paper appears to be most useful when constructing highly accurate artificial boundary conditions (ABCs) for the numerical treatment of problems initially formulated on unbounded domains. Therefore, an effective numerical algorithm for solving such systems becomes an important issue.}, number={4}, journal={Applied Numerical Mathematics}, publisher={Elsevier BV}, author={Ryaben'kii, V.S. and Tsynkov, S.V.}, year={1995}, month={Oct}, pages={489–501} }
 @article{ryaben’kii_tsynkov_1995, title={Artificial Boundary Conditions for the Numerical Solution of External Viscous Flow Problems}, volume={32}, ISSN={0036-1429 1095-7170}, url={http://dx.doi.org/10.1137/0732063}, DOI={10.1137/0732063}, abstractNote={In this paper we describe an algorithm for the nonlocal artificial boundary conditions setting at the external boundary of a computational domain while numerically solving unbounded viscous compressible flow problems past the finite bodies. Our technique is based on the usage of generalized Calderon projection operators and the application of the difference potentials method. Some computational results are presented.}, number={5}, journal={SIAM Journal on Numerical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Ryaben’kii, V. S. and Tsynkov, S. V.}, year={1995}, month={Oct}, pages={1355–1389} }
 @inproceedings{tsynkov_1995, place={Lake Tahoe, Nevada}, title={Nonlocal Artificial Boundary Conditions
               Based on the Difference Potentials Method}, volume={IV}, url={https://www.tib.eu/en/search/id/TIBKAT%3A228255147/A-collection-of-technical-papers-Sixth-International/}, booktitle={Sixth International Symposium on
                 Computational Fluid Dynamics. Collection of Technical Papers}, author={Tsynkov, S. V.}, editor={Hafez, M.Editor}, year={1995}, pages={114–119} }
 @inbook{tsynkov_1995, place={Berlin}, title={Nonlocal Artificial Boundary Conditions for Computation of External Viscous Flows}, ISBN={9783642796562 9783642796548}, url={http://dx.doi.org/10.1007/978-3-642-79654-8_174}, DOI={10.1007/978-3-642-79654-8_174}, booktitle={Computational Mechanics ’95}, publisher={Springer Berlin Heidelberg}, author={Tsynkov, Semyon V.}, editor={Atluri, S. N. and Yagawa, G. and Cruse, T. A.Editors}, year={1995}, pages={1065–1070} }
 @article{artificial boundary conditions for the numerical solution of 
              external viscous flow problems. i_1993, note={[in Russian]}, number={45}, journal={Rossijskaya Akad. Nauk Inst. Prikl. Mat. Preprint}, year={1993} }
 @article{artificial boundary conditions for the numerical solution of 
              external viscous flow problems. ii_1993, note={[in Russian]}, number={46}, journal={Rossijskaya Akad. Nauk Inst. Prikl. Mat. Preprint}, year={1993} }
 @article{tsynkov_1991, title={Application of a model of potential flow to the formulation of
              conditions on the outer boundary for Euler equations. I}, url={https://mathscinet.ams.org/mathscinet-getitem?mr=1156343}, note={[in Russian]}, number={40}, journal={Akad. Nauk SSSR Inst. Prikl. Mat. Preprint}, author={Tsynkov, S. V.}, year={1991}, pages={25} }
 @article{sofronov_tsynkov_1991, title={Application of a model of potential flow to the formulation of
              conditions on the outer boundary for Euler equations. II}, url={https://mathscinet.ams.org/mathscinet-getitem?mr=1156344}, note={[in Russian]}, number={41}, journal={Akad. Nauk SSSR Inst. Prikl. Mat. Preprint}, author={Sofronov, I. L. and Tsynkov, S. V.}, year={1991}, pages={27} }
 @article{boundary equations with projectors in composite domains_1991, url={https://mathscinet.ams.org/mathscinet-getitem?mr=1278545}, note={[in Russian]}, number={112}, journal={Akad. Nauk SSSR Inst. Prikl. Mat. Preprint}, year={1991}, pages={20} }
 @article{decomposition algorithms based on boundary equations with
              projectors_1991, url={https://mathscinet.ams.org/mathscinet-getitem?mr=1278546}, note={[in Russian]}, number={113}, journal={Akad. Nauk SSSR Inst. Prikl. Mat. Preprint}, year={1991}, pages={23} }
 @inproceedings{tsynkov_1991, place={Moscow}, title={Exact  Transfer of Boundary Conditions in
               Subsonic  Problems  of  Computational  Gas  Dynamics}, note={[in Russian]}, booktitle={Construction of Algorithms and Solution of Mathematical Physics Problems}, author={Tsynkov, S. V.}, editor={Zabrodin, A. V. and Voskresensky, G. P.Editors}, year={1991}, pages={194–198} }
 @article{elizarova_tsynkov_chetverushkin_1991, place={New York}, title={Kinetic-Consistent Finite-Difference Schemes in Curvilinear Coordinate Systems}, volume={27}, number={7}, journal={Differential Equations}, publisher={Consultants Bureau}, author={Elizarova, T.G. and Tsynkov, S.V. and Chetverushkin, B.N.}, year={1991}, pages={1161–1169} }
 @article{elizarova_tsynkov_chetverushkin_1991, title={Kinetically consistent difference schemes in curvilinear
              coordinate systems}, volume={27}, url={https://mathscinet.ams.org/mathscinet-getitem?mr=1127501}, note={[in Russian]}, number={7}, journal={Differentsiaļprime nye Uravneniya}, author={Elizarova, T. G. and Tsynkov, S. V. and Chetverushkin, B. N.}, year={1991}, pages={1161–1169, 1285} }
 @article{tsynkov_1990, title={Conditions on the exterior boundary of a computational domain
              in subsonic problems of computational gas dynamics}, url={https://mathscinet.ams.org/mathscinet-getitem?mr=1120837}, note={[in Russian]}, number={108}, journal={Akad. Nauk SSSR Inst. Prikl. Mat. Preprint}, author={Tsynkov, S. V.}, year={1990}, pages={26} }
 @article{elizarova_tsynkov_chetverushkin_1990, title={Derivation of  Invariant  Quasihydrodynamic  Equations
              on  the  Basis   of Kinetic Models}, note={[in Russian]}, number={7}, journal={Akad. Nauk SSSR Inst. Prikl. Mat. Preprint}, author={Elizarova, T. G. and Tsynkov, S. V. and Chetverushkin, B. N.}, year={1990} }
 @article{kamenetskiı̆ d. s._tsynkov_1990, title={Numerical generation of conformal grids in the exterior of a
              bounded simply connected domain based on the method of
              difference potentials}, url={https://mathscinet.ams.org/mathscinet-getitem?mr=1086442}, note={[in Russian]}, number={61}, journal={Akad. Nauk SSSR Inst. Prikl. Mat. Preprint}, author={Kamenetskiı̆ D. S. and Tsynkov, S. V.}, year={1990}, pages={21} }
 @article{kamenetskiı̆ d. s._tsynkov_1990, title={On the construction of images of simply connected domains
              realized by solutions of a system of Beltrami equations}, url={https://mathscinet.ams.org/mathscinet-getitem?mr=1157869}, note={[in Russian]}, number={155}, journal={Akad. Nauk SSSR Inst. Prikl. Mat. Preprint}, author={Kamenetskiı̆ D. S. and Tsynkov, S. V.}, year={1990}, pages={18} }
 @article{elizarova_tsynkov_chetverushkin_1989, title={Construction of kinetically consistent difference schemes on
              curvilinear grids}, url={https://mathscinet.ams.org/mathscinet-getitem?mr=1019147}, note={[in Russian]}, number={8}, journal={Akad. Nauk SSSR Inst. Prikl. Mat. Preprint}, author={Elizarova, T. G. and Tsynkov, S. V. and Chetverushkin, B. N.}, year={1989}, pages={24} }