@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{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{albright_epshteyn_medvinsky_xia_2017, title={High-order numerical schemes based on difference potentials for 2D elliptic problems with material interfaces}, volume={111}, ISSN={["1873-5460"]}, url={https://doi.org/10.1016/j.apnum.2016.08.017}, DOI={10.1016/j.apnum.2016.08.017}, 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={Albright, Jason and Epshteyn, Yekaterina and Medvinsky, Michael and Xia, Qing}, year={2017}, month={Jan}, pages={64–91} }
 @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} }
 @inbook{epshteyn_medvinsky_2015, title={On the Solution of the Elliptic Interface Problems by Difference Potentials Method}, ISBN={9783319197999 9783319198002}, ISSN={1439-7358 2197-7100}, url={http://dx.doi.org/10.1007/978-3-319-19800-2_16}, DOI={10.1007/978-3-319-19800-2_16}, abstractNote={Designing numerical methods with high-order accuracy for problems in irregular domains and/or with interfaces is crucial for the accurate solution of many problems with physical and biological applications. The major challenge here is to design an efficient and accurate numerical method that can capture certain properties of analytical solutions in different domains/subdomains while handling arbitrary geometries and complex structures of the domains. Moreover, in general, any standard method (finite-difference, finite-element, etc.) will fail to produce accurate solutions to interface problems due to discontinuities in the model’s parameters/solutions. In this work, we consider Difference Potentials Method (DPM) as an efficient and accurate solver for the variable coefficient elliptic interface problems.}, booktitle={Lecture Notes in Computational Science and Engineering}, publisher={Springer International Publishing}, author={Epshteyn, Yekaterina and Medvinsky, Michael}, year={2015}, pages={197–205} }
 @inproceedings{britt_medvinsky_tsynkov_turkel_2013, place={Moscow. Russia}, title={High Order Numerical Simulation of the Transmission and Scattering of Waves Using the Method of Difference Potentials}, booktitle={Proceedings of the International Conference "Difference Schemes and Applications" in honor of the 90-th Birthday of Prof. V. S. Ryaben'kii}, publisher={Keldysh Institute of Applied Mathematics}, author={Britt, S. and Medvinsky, M. and Tsynkov, S. and Turkel, E.}, year={2013}, month={May}, 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} }
 @phdthesis{medvinsky_2013, title={High order numerical simulation of waves using regular grids and non-conforming interfaces}, school={Tel Aviv University}, author={Medvinsky, M.}, year={2013} }
 @phdthesis{high order numerical simulation of waves using regular grids and non-conforming interfaces_2013, url={http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=ADA617619}, year={2013}, month={Oct} }
 @article{providing easy access to radio networks_2013, url={http://appft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&p=1&u=%2Fnetahtml%2FPTO%2Fsearch-bool.html&r=1&f=G&l=50&co1=AND&d=PG01&s1=%22Providing+easy+access+radio+networks%22.TTL.&OS=TTL/%22Providing+easy+access+to+radio+networks%22&RS=TTL/%22Providing+easy+access+to+radio+networks%22}, year={2013}, month={Jun} }
 @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{medvinsky_turkel_2010, title={On surface radiation conditions for an ellipse}, volume={234}, ISSN={0377-0427}, url={http://dx.doi.org/10.1016/j.cam.2009.08.011}, DOI={10.1016/j.cam.2009.08.011}, abstractNote={We compare several On Surface Radiation Boundary Conditions in two dimensions, for solving the Helmholtz equation exterior to an ellipse. We also introduce a new boundary condition for an ellipse based on a modal expansion in Mathieu functions. We compare the OSRC to a finite difference method.}, number={6}, journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier BV}, author={Medvinsky, M. and Turkel, E.}, year={2010}, month={Jul}, pages={1647–1655} }
 @article{on surface radiation conditions for an ellipse_2010, url={http://www.sciencedirect.com/science/article/pii/S0377042709004865}, year={2010}, month={Jul} }
 @misc{expiditing seamless roaming in heterogenous networking_2008, url={https://patentscope.wipo.int/search/en/detail.jsf?docId=WO2008073438}, year={2008}, month={Jun} }
 @article{medvinsky_turkel_hetmaniuk_2008, title={Local absorbing boundary conditions for elliptical shaped boundaries}, volume={227}, ISSN={0021-9991}, url={http://dx.doi.org/10.1016/j.jcp.2008.05.010}, DOI={10.1016/j.jcp.2008.05.010}, abstractNote={We compare several local absorbing boundary conditions for solving the Helmholtz equation, by a finite difference or finite element method, exterior to a general scatterer. These boundary conditions are imposed on an artificial elliptical or prolate spheroid outer surface. In order to compare the computational solution with an analytical solution, we consider, as an example, scattering about an ellipse. We solve the Helmholtz equation with both finite differences and finite elements. We also introduce a new boundary condition for an ellipse based on a modal expansion.}, number={18}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Medvinsky, M. and Turkel, E. and Hetmaniuk, U.}, year={2008}, month={Sep}, pages={8254–8267} }
 @misc{masking changes for seamless roaming in heterogenous networking_2008, url={https://patentscope.wipo.int/search/en/detail.jsf?docId=WO2008073492}, year={2008}, month={Jun} }
 @article{ providing easy access to radio networks _2007, url={http://appft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&p=1&u=%2Fnetahtml%2FPTO%2Fsearch-bool.html&r=1&f=G&l=50&co1=AND&d=PG01&s1=%22Providing+easy+access+radio+networks%22.TTL.&OS=TTL/%22Providing+easy+access+to+radio+networks%22&RS=TTL/%22Providing+easy+access+to+radio+networks%22}, year={2007}, month={May} }
 @phdthesis{medvinsky_2007, title={Comparison of local absorbing radiation conditions for scattering about elliptical body}, school={Tel Aviv University}, author={Medvinsky, M.}, year={2007} }
 @phdthesis{comparison of local absorbing radiation conditions for scattering about elliptical body_2007, url={http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.728.1981&rep=rep1&type=pdf}, year={2007} }
 @inbook{on the solution of the elliptic interface problems by difference potentials method }