@article{chertock_kurganov_redle_wu_2024, title={A NEW LOCALLY DIVERGENCE-FREE PATH-CONSERVATIVE CENTRAL-UPWIND SCHEME FOR IDEAL AND SHALLOW WATER MAGNETOHYDRODYNAMICS}, volume={46}, ISSN={["1095-7197"]}, DOI={10.1137/22M1539009}, number={3}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Chertock, Alina and Kurganov, Alexander and Redle, Michael and Wu, Kailiang}, year={2024}, pages={A1998–A2024} }
@article{chertock_chu_kurganov_2024, title={Accurate deterministic projection methods for stiff detonation waves}, url={http://dx.doi.org/10.4310/cms.2024.v22.n4.a1}, DOI={10.4310/cms.2024.v22.n4.a1}, journal={Communications in Mathematical Sciences}, author={Chertock, Alina and Chu, Shaoshuai and Kurganov, Alexander}, year={2024} }
@article{chertock_kurganov_redle_zeitlin_2024, title={Locally divergence-free well-balanced path-conservative central-upwind schemes for rotating shallow water MHD}, volume={518}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2024.113300}, abstractNote={We develop a new second-order flux globalization based path-conservative central-upwind (PCCU) scheme for rotating shallow water magnetohydrodynamic equations. The new scheme is designed not only to maintain the divergence-free constraint of the magnetic field at the discrete level but also to satisfy the well-balanced (WB) property by exactly preserving some physically relevant steady states of the underlying system. The locally divergence-free constraint of the magnetic field is enforced by following the method recently introduced in Chertock et al. (2024) [19]: we consider a Godunov-Powell modified version of the studied system, introduce additional equations by spatially differentiating the magnetic field equations, and modify the reconstruction procedures for magnetic field variables. The WB property is ensured by implementing a flux globalization approach within the PCCU scheme, leading to a method capable of preserving both still- and moving-water equilibria exactly. In addition to provably achieving both the WB and divergence-free properties, the new method is implemented on an unstaggered grid and does not require any (approximate) Riemann problem solvers. The performance of the proposed method is demonstrated in several numerical experiments that confirms robustness, a high resolution of obtained results, and a lack of spurious oscillations.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Chertock, Alina and Kurganov, Alexander and Redle, Michael and Zeitlin, Vladimir}, year={2024}, month={Dec} }
@article{chertock_herty_iskhakov_janajra_kurganov_lukacova-medvid'ova_2024, title={New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties}, volume={6}, ISSN={["2661-8893"]}, DOI={10.1007/s42967-024-00392-z}, journal={COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION}, author={Chertock, Alina and Herty, Michael and Iskhakov, Arsen S. and Janajra, Safa and Kurganov, Alexander and Lukacova-Medvid'ova, Maria}, year={2024}, month={Jun} }
@article{chertock_leonard_2024, title={Simulating Partial Differential Equations with Neural Networks}, volume={35}, ISBN={["978-3-031-55266-3", "978-3-031-55263-2"]}, ISSN={["2199-305X"]}, url={http://dx.doi.org/10.1007/978-3-031-55264-9_4}, DOI={10.1007/978-3-031-55264-9_4}, journal={HYPERBOLIC PROBLEMS: THEORY, NUMERICS, APPLICATIONS, VOL II, HYP2022}, author={Chertock, Anna and Leonard, Christopher}, year={2024}, pages={39–49} }
@article{wang_chertock_cui_kurganov_zhang_2023, title={A diffuse-domain-based numerical method for a chemotaxis-fluid model}, volume={2}, ISSN={["1793-6314"]}, DOI={10.1142/S0218202523500094}, abstractNote={ In this paper, we consider a coupled chemotaxis-fluid system that models self-organized collective behavior of oxytactic bacteria in a sessile drop. This model describes the biological chemotaxis phenomenon in the fluid environment and couples a convective chemotaxis system for the oxygen-consuming and oxytactic bacteria with the incompressible Navier–Stokes equations subject to a gravitational force, which is proportional to the relative surplus of the cell density compared to the water density. We develop a new positivity preserving and high-resolution method for the studied chemotaxis-fluid system. Our method is based on the diffuse-domain approach, which we use to derive a new chemotaxis-fluid diffuse-domain (cf-DD) model for simulating bioconvection in complex geometries. The drop domain is imbedded into a larger rectangular domain, and the original boundary is replaced by a diffuse interface with finite thickness. The original chemotaxis-fluid system is reformulated on the larger domain with additional source terms that approximate the boundary conditions on the physical interface. We show that the cf-DD model converges to the chemotaxis-fluid model asymptotically as the width of the diffuse interface shrinks to zero. We numerically solve the resulting cf-DD system by a second-order hybrid finite-volume finite-difference method and demonstrate the performance of the proposed approach on a number of numerical experiments that showcase several interesting chemotactic phenomena in sessile drops of different shapes, where the bacterial patterns depend on the droplet geometries. }, journal={MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES}, author={Wang, Chenxi and Chertock, Alina and Cui, Shumo and Kurganov, Alexander and Zhang, Zhen}, year={2023}, month={Feb} }
@article{chertock_chu_kurganov_2023, title={Adaptive High-Order A-WENO Schemes Based on a New Local Smoothness Indicator}, volume={4}, ISSN={["2079-7370"]}, DOI={10.4208/eajam.2022-313.160123}, abstractNote={We develop new adaptive alternative weighted essentially non-oscillatory (A-WENO) schemes for hyperbolic systems of conservation laws.The new schemes employ the recently proposed local characteristic decomposition based central-upwind numerical fluxes, the three-stage third-order strong stability preserving Runge-Kutta time integrator, and the fifth-order WENO-Z interpolation.The adaptive strategy is implemented by applying the limited interpolation only in the parts of the computational domain where the solution is identified as rough with the help of a smoothness indicator.We develop and use a new simple and robust local smoothness indicator (LSI), which is applied to the solutions computed at each of the three stages of the ODE solver.The new LSI and adaptive A-WENO schemes are tested on the Euler equations of gas dynamics.We implement the proposed LSI using the pressure, which remains smooth at contact discontinuities, while our goal is to detect other rough areas and apply the limited interpolation mostly in the neighborhoods of the shock waves.We demonstrate that the new adaptive schemes are highly accurate, non-oscillatory, and robust.They outperform their fully limited counterparts (the A-WENO schemes with the same numerical fluxes and ODE solver but with the WENO-Z interpolation employed everywhere) while being less computationally expensive.}, journal={EAST ASIAN JOURNAL ON APPLIED MATHEMATICS}, author={Chertock, Alina and Chu, Shaoshuai and Kurganov, Alexander}, year={2023}, month={Apr} }
@article{chertock_chu_kurganov_2023, title={Adaptive High-Order A-WENO Schemes Based on a New Local Smoothness Indicator}, volume={13}, ISSN={["2079-7370"]}, DOI={10.4208/eajam.2022-313.160123August2023}, number={3}, journal={EAST ASIAN JOURNAL ON APPLIED MATHEMATICS}, author={Chertock, Alina and Chu, Shaoshuai and Kurganov, Alexander}, year={2023}, month={Aug}, pages={576–609} }
@article{chertock_mathematics_leonard_tsynkov_utyuzhnikov_mechanical_2023, title={Denoising convolution algorithms and applications to SAR signal processing}, volume={1}, url={http://dx.doi.org/10.3934/cac.2023008}, DOI={10.3934/cac.2023008}, abstractNote={Convolutions are one of the most important operations in signal processing. They often involve large arrays and require significant computing time. Moreover, in practice, the signal data to be processed by convolution may be corrupted by noise. In this paper, we introduce a new method for computing the convolutions in the quantized tensor train (QTT) format and removing noise from data using the QTT decomposition. We demonstrate the performance of our method using a common mathematical model for synthetic aperture radar (SAR) processing that involves a sinc kernel and present the entire cost of decomposing the original data array, computing the convolutions, and then reformatting the data back into full arrays.}, number={2}, journal={Communications on Analysis and Computation}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={Chertock, Alina and Mathematics, North Carolina State University and Leonard, Chris and Tsynkov, Semyon and Utyuzhnikov, Sergey and Mechanical, Aerospace}, year={2023}, pages={135–156} }
@article{chertock_kurganov_lukáčová-medvid'ová_spichtinger_wiebe_2023, title={Stochastic Galerkin method for cloud simulation. Part II: A fully random Navier-Stokes-cloud model}, volume={479}, url={http://dx.doi.org/10.1016/j.jcp.2023.111987}, DOI={10.1016/j.jcp.2023.111987}, abstractNote={This paper is a continuation of the work presented in Chertock et al. (2019) [8]. We study uncertainty propagation in warm cloud dynamics of weakly compressible fluids. The mathematical model is governed by a multiscale system of PDEs in which the macroscopic fluid dynamics is described by a weakly compressible Navier-Stokes system and the microscopic cloud dynamics is modeled by a convection-diffusion-reaction system. In order to quantify uncertainties present in the system, we derive and implement a generalized polynomial chaos stochastic Galerkin method. Unlike the first part of this work, where we restricted our consideration to the partially stochastic case in which the uncertainties were only present in the cloud physics equations, we now study a fully random Navier-Stokes-cloud model in which we include randomness in the macroscopic fluid dynamics as well. We conduct a series of numerical experiments illustrating the accuracy and efficiency of the developed approach.}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Chertock, A. and Kurganov, A. and Lukáčová-Medvid'ová, M. and Spichtinger, P. and Wiebe, B.}, year={2023}, month={Apr}, pages={111987} }
@article{chertock_chu_herty_kurganov_lukacova-medvid'ova_2023, title={Local characteristic decomposition based central-upwind scheme}, volume={473}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2022.111718}, abstractNote={We propose novel less diffusive schemes for conservative one- and two-dimensional hyperbolic systems of nonlinear partial differential equations (PDEs). The main challenges in the development of accurate and robust numerical methods for the studied systems come from the complicated wave structures, such as shocks, rarefactions and contact discontinuities, arising even for smooth initial conditions. In order to reduce the diffusion in the original central-upwind schemes, we use a local characteristic decomposition procedure to develop a new class of central-upwind schemes. We apply the developed schemes to the one- and two-dimensional Euler equations of gas dynamics to illustrate the performance on a variety of examples. The obtained numerical results clearly demonstrate that the proposed new schemes outperform the original central-upwind schemes.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Chertock, Alina and Chu, Shaoshuai and Herty, Michael and Kurganov, Alexander and Lukacova-Medvid'ova, Maria}, year={2023}, month={Jan} }
@article{chertock_degond_dimarco_lukáčová-medvid’ová_ruhi_2022, title={On a hybrid continuum-kinetic model for complex fluids}, volume={3}, url={http://dx.doi.org/10.1007/s42985-022-00198-9}, DOI={10.1007/s42985-022-00198-9}, abstractNote={AbstractIn the present work, we first introduce a general framework for modelling complex multiscale fluids and then focus on the derivation and analysis of a new hybrid continuum-kinetic model. In particular, we combine conservation of mass and momentum for an isentropic macroscopic model with a kinetic representation of the microscopic behavior. After introducing a small scale of interest, we compute the complex stress tensor by means of the Irving-Kirkwood formula. The latter requires an expansion of the kinetic distribution around an equilibrium state and a successive homogenization over the fast in time and small in space scale dynamics. For a new hybrid continuum-kinetic model the results of linear stability analysis indicate a conditional stability in the relevant low speed regimes and linear instability for high speed regimes for higher modes. Extensive numerical experiments confirm that the proposed multiscale model can reflect new phenomena of complex fluids not being present in standard Newtonian fluids. Consequently, the proposed general technique can be successfully used to derive new interesting systems combining the macro and micro structure of a given physical problem.}, number={5}, journal={Partial Differential Equations and Applications}, publisher={Springer Science and Business Media LLC}, author={Chertock, A. and Degond, P. and Dimarco, G. and Lukáčová-Medvid’ová, M. and Ruhi, A.}, year={2022}, month={Oct} }
@article{chertock_kurganov_wu_yan_2023, title={Well-balanced numerical method for atmospheric flow equations with gravity}, volume={439}, ISSN={["1873-5649"]}, DOI={10.1016/j.amc.2022.127587}, abstractNote={We are interested in simulating gravitationally stratified atmospheric flows governed by the compressible Euler equations in irregular domains. In such simulations, one of the challenges arises when the computations are conducted on a Cartesian grid. The use of regular rectangular grids that intersect with the irregular boundaries leads to the generation of arbitrarily small and highly distorted computational cells adjacent to the boundaries of the domain. The appearance of such cells may affect both the stability and efficiency of the numerical method and therefore require special attention. In order to overcome this difficulty, we introduce a structured quadrilateral mesh, which is designed for the irregular domain at hand, and solve the studied atmospheric flow equations using a second-order central-upwind scheme. In addition, the resulting numerical method is developed to provide a well-balanced discretization of the underlying system. The latter is achieved by rewriting the governing equations in terms of equilibrium variables representing perturbations of the known background equilibrium state. The proposed method is tested in a number of numerical experiments, including the buoyant bubble rising and interacting with an (zeppelin) obstacle and the Lee wave generation due to topography. The obtained numerical results demonstrate high resolution and robustness of the proposed computational approach.}, journal={APPLIED MATHEMATICS AND COMPUTATION}, author={Chertock, Alina and Kurganov, Alexander and Wu, Tong and Yan, Jun}, year={2023}, month={Feb} }
@article{chertock_chu_kurganov_2021, title={Hybrid Multifluid Algorithms Based on the Path-Conservative Central-Upwind Scheme}, volume={89}, ISSN={["1573-7691"]}, DOI={10.1007/s10915-021-01656-z}, number={2}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Chertock, Alina and Chu, Shaoshuai and Kurganov, Alexander}, year={2021}, month={Nov} }
@article{chertock_kurganov_liu_liu_wu_2022, title={Well-Balancing via Flux Globalization: Applications to Shallow Water Equations with Wet/Dry Fronts}, volume={90}, ISSN={["1573-7691"]}, DOI={10.1007/s10915-021-01680-z}, number={1}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Chertock, Alina and Kurganov, Alexander and Liu, Xin and Liu, Yongle and Wu, Tong}, year={2022}, month={Jan} }
@inbook{chertock_kurganov_miller_yan_2020, title={Central-upwind scheme for a non-hydrostatic Saint-Venant
system}, volume={10}, booktitle={Hyperbolic problems: theory, numerics, applications}, publisher={Am. Inst. Math. Sci. (AIMS), Springfield, MO}, author={Chertock, Alina and Kurganov, Alexander and Miller, Jason and Yan, Jun}, year={2020}, pages={25–41} }
@article{chertock_kurganov_wu_2020, title={Operator splitting based central-upwind schemes for shallow water equations with moving bottom topography}, volume={18}, DOI={10.4310/cms.2020.v18.n8.a3}, abstractNote={. In this paper, we develop a robust and eﬃcient numerical method for shallow water equations with moving bottom topography. The model consists of the Saint-Venant system governing the water ﬂow coupled with the Exner equation for the sediment transport. One of the main diﬃculties in designing good numerical methods for such models is related to the fact that the speed of water surface gravity waves is typically much faster than the speed at which the changes in the bottom topography occur. This imposes a severe stability restriction on the size of time steps, which, in turn, leads to excessive numerical diﬀusion that aﬀects the computed bottom structure. In order to overcome this diﬃculty, we develop an operator splitting approach for the underlying coupled system, which allows one to treat slow and fast waves in a diﬀerent manner and using diﬀerent time steps. Our method is based on the application of a ﬁnite-volume central-upwind scheme introduced in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5:133–160, 2007], and incorporates a staggered grid strategy needed for a proper approximation of the bottom topography function. A number of one-and two-dimensional numerical examples are presented to demonstrate the performance of the proposed method.}, number={8}, journal={Communications in Mathematical Sciences}, publisher={International Press of Boston}, author={Chertock, Alina and Kurganov, Alexander and Wu, Tong}, year={2020}, pages={2149–2168} }
@article{cheng_chertock_herty_kurganov_wu_2019, title={A New Approach for Designing Moving-Water Equilibria Preserving Schemes for the Shallow Water Equations}, volume={80}, ISSN={["1573-7691"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1007/s10915-019-00947-w}, DOI={10.1007/s10915-019-00947-w}, number={1}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Cheng, Yuanzhen and Chertock, Alina and Herty, Michael and Kurganov, Alexander and Wu, Tong}, year={2019}, month={Jul}, pages={538–554} }
@article{chertock_kurganov_ricchiuto_wu_2019, title={Adaptive moving mesh upwind scheme for the two-species chemotaxis model}, volume={77}, ISSN={0898-1221}, url={http://dx.doi.org/10.1016/j.camwa.2019.01.021}, DOI={10.1016/j.camwa.2019.01.021}, abstractNote={Chemotaxis systems are used to model the propagation, aggregation and pattern formation of bacteria/cells in response to an external stimulus, usually a chemical one. A common property of all chemotaxis systems is their ability to model a concentration phenomenon—rapid growth of the cell density in small neighborhoods of concentration points/curves. More precisely, the solution may develop singular, spiky structures, or even blow up in finite time. Therefore, the development of accurate and computationally efficient numerical methods for the chemotaxis models is a challenging task. We study the two-species Patlak–Keller–Segel type chemotaxis system, in which the two species do not compete, but have different chemotactic sensitivities, which may lead to a significantly difference in cell density growth rates. This phenomenon was numerically investigated in Kurganov and Lukáčová-Medviďová (2014) and Chertock et al. (2018), where second- and higher-order methods on uniform Cartesian grids were developed. However, in order to achieve high resolution of the density spikes developed by the species with a lower chemotactic sensitivity, a very fine mesh had to be utilized and thus the efficiency of the numerical method was affected. In this work, we consider an alternative approach relying on mesh adaptation, which helps to improve the approximation of the singular structures evolved by chemotaxis models. We develop, in particular, an adaptive moving mesh (AMM) finite-volume semi-discrete upwind method for the two-species chemotaxis system. The proposed AMM technique allows one to increase the density of mesh nodes at the blowup regions. This helps to substantially improve the resolution while using a relatively small number of finite-volume cells.}, number={12}, journal={Computers & Mathematics with Applications}, publisher={Elsevier BV}, author={Chertock, Alina and Kurganov, Alexander and Ricchiuto, Mario and Wu, Tong}, year={2019}, month={Jun}, pages={3172–3185} }
@article{liu_chertock_kurganov_2019, title={An asymptotic preserving scheme for the two-dimensional shallow water equations with Coriolis forces}, volume={391}, ISSN={["1090-2716"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1016/j.jcp.2019.04.035}, DOI={10.1016/j.jcp.2019.04.035}, abstractNote={We consider the two-dimensional Saint-Venant system of shallow water equations with Coriolis forces. We focus on the case of a low Froude number, in which the system is stiff and conventional explicit numerical methods are extremely inefficient and often impractical. Our goal is to design an asymptotic preserving (AP) scheme, which is uniformly asymptotically consistent and stable for a broad range of (low) Froude numbers. The goal is achieved using the flux splitting proposed in [Haack et al., Commun. Comput. Phys., 12 (2012), pp. 955–980] in the context of isentropic Euler and Navier-Stokes equations. We split the flux into the stiff and nonstiff parts and then use an implicit-explicit approach: apply an explicit hyperbolic solver (we use the second-order central-upwind scheme) to the nonstiff part of the system while treating the stiff part of it implicitly. Moreover, the stiff part of the flux is linear and therefore we reduce the implicit stage of the proposed method to solving a Poisson-type elliptic equation, which is discretized using a standard second-order central difference scheme. We conduct a series of numerical experiments, which demonstrate that the developed AP scheme achieves the theoretical second-order rate of convergence and the time-step stability restriction is independent of the Froude number. This makes the proposed AP scheme an efficient and robust alternative to fully explicit numerical methods.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Liu, Xin and Chertock, Alina and Kurganov, Alexander}, year={2019}, month={Aug}, pages={259–279} }
@article{chertock_kurganov_liu_2020, title={Finite-Volume-Particle Methods for the Two-Component Camassa-Holm System}, volume={27}, ISSN={["1991-7120"]}, DOI={10.4208/cicp.OA-2018-0325}, abstractNote={We study the two-component Camassa-Holm (2CH) equations as a model for the long time water wave propagation. Compared with the classical Saint-Venant system, it has the advantage of preserving the waves amplitude and shape for a long time. We present two different numerical methods—finite volume (FV) and hybrid finite-volume-particle (FVP) ones. In the FV setup, we rewrite the 2CH equations in a conservative form and numerically solve it by the central-upwind scheme, while in the FVP method, we apply the central-upwind scheme to the density equation only while solving the momentum and velocity equations by a deterministic particle method. Numerical examples are shown to verify the accuracy of both FV and FVP methods. The obtained results demonstrate that the FVP method outperforms the FV method and achieves a superior resolution thanks to a low-diffusive nature of a particle approximation. AMS subject classifications: 65M08, 76M12, 76M28, 86-08, 76M25, 35L65}, number={2}, journal={COMMUNICATIONS IN COMPUTATIONAL PHYSICS}, author={Chertock, Alina and Kurganov, Alexander and Liu, Yongle}, year={2020}, month={Feb}, pages={480–502} }
@article{chertock_kurganov_2019, title={High-Resolution Positivity and Asymptotic Preserving Numerical Methods for Chemotaxis and Related Models}, ISBN={["978-3-030-20296-5"]}, ISSN={["2164-3725"]}, DOI={10.1007/978-3-030-20297-2_4}, abstractNote={Many microorganisms exhibit a special pattern formation at the presence of a chemoattractant, food, light, or areas with high oxygen concentration. Collective cell movement can be described by a system of nonlinear PDEs on both macroscopic and microscopic levels. The classical PDE chemotaxis model is the Patlak-Keller-Segel system, which consists of a convection-diffusion equation for the cell density and a reaction-diffusion equation for the chemoattractant concentration. At the cellular (microscopic) level, a multiscale chemotaxis models can be used. These models are based on a combination of the macroscopic evolution equation for chemoattractant and microscopic models for cell evolution. The latter is governed by a Boltzmann-type kinetic equation with a local turning kernel operator that describes the velocity change of the cells. A common property of the chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in solutions rapidly growing in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. In either case, capturing such singular solutions numerically is a challenging problem and the use of higher-order methods and/or adaptive strategies is often necessary. In addition, positivity preserving is an absolutely crucial property a good numerical method used to simulate chemotaxis should satisfy: this is the only way to guarantee a nonlinear stability of the method. For kinetic chemotaxis systems, it is also essential that numerical methods provide a consistent and stable discretization in certain asymptotic regimes. In this paper, we review some of the recent advances in developing of high-resolution finite-volume and finite-difference numerical methods that possess the aforementioned properties of the chemotaxis-type systems.}, journal={ACTIVE PARTICLES, VOL 2: ADVANCES IN THEORY, MODELS, AND APPLICATIONS}, author={Chertock, Alina and Kurganov, Alexander}, year={2019}, pages={109–148} }
@article{chertock_degond_hecht_vincent_2019, title={Incompressible limit of a continuum model of tissue growth with segregation for two cell populations}, volume={16}, ISSN={["1551-0018"]}, url={http://dx.doi.org/10.3934/mbe.2019290}, DOI={10.3934/mbe.2019290}, abstractNote={This paper proposes a model for the growth of two interacting populations of cells that do not mix. The dynamics is driven by pressure and cohesion forces on the one hand and proliferation on the other hand. Contrasting with earlier works which assume that the two populations are initially segregated, our model can deal with initially mixed populations as it explicitly incorporates a repul-sion force that enforces segregation. To balance segregation instabilities potentially triggered by the repulsion force, our model also incorporates a fourth order diffusion. In this paper, we study the influ-ence of the model parameters thanks to one-dimensional simulations using a finite-volume method for a relaxation approximation of the fourth order diffusion. Then, following earlier works on the single population case, we provide formal arguments that the model approximates a free boundary Hele Shaw type model that we characterise using both analytical and numerical arguments.}, number={5}, journal={MATHEMATICAL BIOSCIENCES AND ENGINEERING}, author={Chertock, Alina and Degond, Pierre and Hecht, Sophie and Vincent, Jean-Paul}, year={2019}, pages={5804–5835} }
@article{chertock_degond_hecht_vincent_2019, title={Incompressible limit of a continuum model of tissue growth with segregation for two cell populations}, volume={16}, url={https://doi.org/10.3934/mbe.2019290}, DOI={https://doi-org.prox.lib.ncsu.edu/10.1007/s10915-019-00947-w}, number={5}, journal={Math. Biosci. Eng.}, author={Chertock, Alina and Degond, Pierre and Hecht, Sophie and Vincent, Jean-Paul}, year={2019}, pages={5804–5835} }
@article{one-dimensional/two-dimensional coupling approach with quadrilateral confluence region for modeling river systems_2019, url={http://dx.doi.org/10.1007/s10915-019-00985-4}, DOI={10.1007/s10915-019-00985-4}, journal={Journal of Scientific Computing}, year={2019}, month={Jun} }
@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} }
@article{chertock_kurganov_lukáčová-medvid’ová_spichtinger_wiebe_2019, title={Stochastic Galerkin method for cloud simulation}, volume={5}, ISSN={2353-6438}, url={http://dx.doi.org/10.1515/mcwf-2019-0005}, DOI={10.1515/mcwf-2019-0005}, abstractNote={AbstractWe develop a stochastic Galerkin method for a coupled Navier-Stokes-cloud system that models dynamics of warm clouds. Our goal is to explicitly describe the evolution of uncertainties that arise due to unknown input data, such as model parameters and initial or boundary conditions. The developed stochastic Galerkin method combines the space-time approximation obtained by a suitable finite volume method with a spectral-type approximation based on the generalized polynomial chaos expansion in the stochastic space. The resulting numerical scheme yields a second-order accurate approximation in both space and time and exponential convergence in the stochastic space. Our numerical results demonstrate the reliability and robustness of the stochastic Galerkin method. We also use the proposed method to study the behavior of clouds in certain perturbed scenarios, for examples, the ones leading to changes in macroscopic cloud pattern as a shift from hexagonal to rectangular structures.}, number={1}, journal={Mathematics of Climate and Weather Forecasting}, publisher={Walter de Gruyter GmbH}, author={Chertock, A. and Kurganov, A. and Lukáčová-Medvid’ová, M. and Spichtinger, P. and Wiebe, B.}, year={2019}, month={Jan}, pages={65–106} }
@article{chertock_coco_kurganov_russo_2018, title={A second-order finite-difference method for compressible
fluids in domains with moving boundaries}, volume={23}, number={1}, journal={Commun. Comput. Phys.}, author={Chertock, Alina and Coco, Armando and Kurganov, Alexander and Russo, Giovanni}, year={2018}, pages={230–263} }
@article{chertock_tan_yan_2018, title={AN ASYMPTOTIC PRESERVING SCHEME FOR KINETIC MODELS WITH SINGULAR LIMIT}, volume={11}, ISSN={["1937-5077"]}, url={https://doi-org.prox.lib.ncsu.edu/10.3934/krm.2018030}, DOI={10.3934/krm.2018030}, abstractNote={We propose a new class of asymptotic preserving schemes to solve kinetic equations with mono-kinetic singular limit. The main idea to deal with the singularity is to transform the equations by appropriate scalings in velocity. In particular, we study two biologically related kinetic systems. We derive the scaling factors and prove that the rescaled solution does not have a singular limit, under appropriate spatial non-oscillatory assumptions, which can be verified numerically by a newly developed asymptotic preserving scheme. We set up a few numerical experiments to demonstrate the accuracy, stability, efficiency and asymptotic preserving property of the schemes.}, number={4}, journal={KINETIC AND RELATED MODELS}, author={Chertock, Alina and Tan, Changhui and Yan, Bokai}, year={2018}, month={Aug}, pages={735–756} }
@article{chertock_kurganov_lukáčová-medvid’ová_nur özcan_2019, title={An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions}, volume={12}, ISSN={1937-5077}, url={http://dx.doi.org/10.3934/krm.2019009}, DOI={10.3934/krm.2019009}, abstractNote={In this paper, we study two-dimensional multiscale chemotaxis models based on a combination of the macroscopic evolution equation for chemoattractant and microscopic models for cell evolution. The latter is governed by a Boltzmann-type kinetic equation with a local turning kernel operator which describes the velocity change of the cells. The parabolic scaling yields a non-dimensional kinetic model with a small parameter, which represents the mean free path of the cells. We propose a new asymptotic preserving numerical scheme that reflects the convergence of the studied micro-macro model to its macroscopic counterpart-the Patlak-Keller-Segel system-in the singular limit. The method is based on the operator splitting strategy and a suitable combination of the higher-order implicit and explicit time discretizations. In particular, we use the so-called even-odd decoupling and approximate the stiff terms arising in the singular limit implicitly. We prove that the resulting scheme satisfies the asymptotic preserving property. More precisely, it yields a consistent approximation of the Patlak-Keller-Segel system as the mean-free path tends to 0. The derived asymptotic preserving method is used to get better insight to the blowup behavior of two-dimensional kinetic chemotaxis model.}, number={1}, journal={Kinetic & Related Models}, publisher={American Institute of Mathematical Sciences (AIMS)}, author={Chertock, Alina and Kurganov, Alexander and Lukáčová-Medvid’ová, Mária and Nur Özcan, Șeyma}, year={2019}, pages={195–216} }
@inbook{chertock_herty_özcan_2018, title={Well-Balanced Central-Upwind Schemes for 2x2 Systems of Balance Laws}, volume={236}, ISBN={9783319915449 9783319915456}, ISSN={2194-1009 2194-1017}, url={http://dx.doi.org/10.1007/978-3-319-91545-6_28}, DOI={10.1007/978-3-319-91545-6_28}, abstractNote={In this study, we have developed a well-balanced second-order central-upwind scheme for $$2\times 2$$ systems of balance laws, in particular, the models of isothermal gas dynamics with source and traffic flow with relaxation to equilibrium velocities. The new scheme is based on modifications in the reconstruction and evolution steps of a Godunov-type central-upwind method. The first step of this modification is to introduce an equilibrium variable obtained from incorporating the source term into the flux. By reconstructing equilibrium variables and using them in the well-balanced evolution process, we have illustrated that the proposed scheme being well balanced, namely, it preserves steady states of the system.}, booktitle={Theory, Numerics and Applications of Hyperbolic Problems I}, publisher={Springer International Publishing}, author={Chertock, Alina and Herty, Michael and Özcan, Şeyma Nur}, year={2018}, pages={345–361} }
@article{chertock_cui_kurganov_ozcan_tadmor_2018, title={Well-balanced schemes for the Euler equations with gravitation: Conservative formulation using global fluxes}, volume={358}, ISSN={["1090-2716"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1016/j.jcp.2017.12.026}, DOI={10.1016/j.jcp.2017.12.026}, abstractNote={We develop a second-order well-balanced central-upwind scheme for the compressible Euler equations with gravitational source term. Here, we advocate a new paradigm based on a purely conservative reformulation of the equations using global fluxes. The proposed scheme is capable of exactly preserving steady-state solutions expressed in terms of a nonlocal equilibrium variable. A crucial step in the construction of the second-order scheme is a well-balanced piecewise linear reconstruction of equilibrium variables combined with a well-balanced central-upwind evolution in time, which is adapted to reduce the amount of numerical viscosity when the flow is at (near) steady-state regime. We show the performance of our newly developed central-upwind scheme and demonstrate importance of perfect balance between the fluxes and gravitational forces in a series of one- and two-dimensional examples.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Chertock, Alina and Cui, Shumo and Kurganov, Alexander and Ozcan, Seyma Nur and Tadmor, Eitan}, year={2018}, month={Apr}, pages={36–52} }
@inbook{chertock_2017, title={A Practical Guide to Deterministic Particle Methods}, volume={18}, ISBN={9780444639103}, ISSN={1570-8659}, url={http://dx.doi.org/10.1016/bs.hna.2016.11.004}, DOI={10.1016/bs.hna.2016.11.004}, abstractNote={The past several decades have seen significant development in the design and numerical analysis of particle methods for approximating solutions of PDEs. In these methods, a numerical solution is sought as a linear combination of Dirac delta-functions located at certain points. The locations and coefficients (weights) of the delta-functions are first chosen to accurately approximate the initial data and then are evolved in time according to the system of ODEs obtained from a weak formulation of the considered problem. The main advantage of the particle methods is their low numerical diffusion that allows them to capture a variety of nonlinear waves with a high resolution. Even though the most "natural" application of the particle methods is linear transport equations, over the years, the range of these methods has been extended for approximating solutions of convection–diffusion and dispersive equations and general nonlinear problems. In this chapter, we provide a mathematical introduction to deterministic particle methods and review different aspects of their practical implementation such as recovering an approximate solution from its particle distribution and an investigation of various particle redistribution algorithms.}, booktitle={Handbook of Numerical Analysis}, publisher={Elsevier}, author={Chertock, A.}, year={2017}, pages={177–202} }
@inbook{cheng_chertock_kurganov_2017, title={A Simple Finite-Volume Method on a Cartesian Mesh for Pedestrian Flows with Obstacles}, volume={199}, ISBN={9783319573960 9783319573977}, ISSN={2194-1009 2194-1017}, url={http://dx.doi.org/10.1007/978-3-319-57397-7_4}, DOI={10.1007/978-3-319-57397-7_4}, abstractNote={We considerCheng, Yuanzhen a two-dimensionalChertock, Alina pedestrianKurganov, Alexander flow model with obstacles governed by scalar hyperbolic conservation laws, in which the flux is implicitly dependent on the density through the Eikonal equation. We propose a simple second-order finite-volume method, which is applicable to the case of obstacles of arbitrary shapes. Though the method is only first-order accurate near the obstacles, it is robust and provides sharp resolution of discontinuities as illustrated in a number of numerical experiments.}, booktitle={Springer Proceedings in Mathematics & Statistics}, publisher={Springer International Publishing}, author={Cheng, Yuanzhen and Chertock, Alina and Kurganov, Alexander}, year={2017}, pages={43–55} }
@article{chertock_degond_neusser_2017, title={An asymptotic-preserving method for a relaxation of the Navier-Stokes-Korteweg equations}, volume={335}, ISSN={["1090-2716"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1016/j.jcp.2017.01.030}, DOI={10.1016/j.jcp.2017.01.030}, abstractNote={The Navier–Stokes–Korteweg (NSK) equations are a classical diffuse-interface model for compressible two-phase flows. As direct numerical simulations based on the NSK system are quite expensive and in some cases even impossible, we consider a relaxation of the NSK system, for which robust numerical methods can be designed. However, time steps for explicit numerical schemes depend on the relaxation parameter and therefore numerical simulations in the relaxation limit are very inefficient. To overcome this restriction, we propose an implicit–explicit asymptotic-preserving finite volume method. We prove that the new scheme provides a consistent discretization of the NSK system in the relaxation limit and demonstrate that it is capable of accurately and efficiently computing numerical solutions of problems with realistic density ratios and small interfacial widths.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Chertock, Alina and Degond, Pierre and Neusser, Jochen}, year={2017}, month={Apr}, pages={387–403} }
@article{chertock_epshteyn_hu_kurganov_2018, title={High-order positivity-preserving hybrid finite-volume-finite-difference methods for chemotaxis systems}, volume={44}, ISSN={["1572-9044"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1007/s10444-017-9545-9}, DOI={10.1007/s10444-017-9545-9}, number={1}, journal={ADVANCES IN COMPUTATIONAL MATHEMATICS}, author={Chertock, Alina and Epshteyn, Yekaterina and Hu, Hengrui and Kurganov, Alexander}, year={2018}, month={Feb}, pages={327–350} }
@article{chertock_cui_kurganov_2017, title={Hybrid Finite-Volume-Particle Method for Dusty Gas Flows}, volume={3}, ISSN={2426-8399}, url={http://dx.doi.org/10.5802/smai-jcm.23}, DOI={10.5802/smai-jcm.23}, abstractNote={We first study the one-dimensional dusty gas flow modeled by the two-phase system composed of a gaseous carrier (gas phase) and a particulate suspended phase (dust phase). The gas phase is modeled by the compressible Euler equations of gas dynamics and the dust phase is modeled by the pressureless gas dynamics equations. These two sets of conservation laws are coupled through source terms that model momentum and heat transfers between the phases. When an Eulerian method is adopted for this model, one can notice the obtained numerical results are typically significantly affected by numerical diffusion. This phenomenon occurs since the pressureless gas equations are nonstrictly hyperbolic and have degenerate structure in which singular delta shocks are formed, and these strong singularities are vulnerable to the numerical diffusion. We introduce a low dissipative hybrid finite-volume-particle method in which the compressible Euler equations for the gas phase are solved by a central-upwind scheme, while the pressureless gas dynamics equations for the dust phase are solved by a sticky particle method. The obtained numerical results demonstrate that our hybrid method provides a sharp resolution even when a relatively small number of particle is used. We then extend the hybrid finite-volume-particle method to the three-dimensional dusty gas flows with axial symmetry. In the studied model, gravitational effects are taken into account. This brings an additional level of complexity to the development of the finite-volume-particle method since a delicate balance between the flux and gravitational source terms should be respected at the discrete level. We test the proposed method on a number of numerical examples including the one that models volcanic eruptions. Math. classification. 65M08, 76M12, 76M28, 86-08, 76M25, 35L65.}, journal={SMAI Journal of Computational Mathematics}, publisher={Cellule MathDoc/CEDRAM}, author={Chertock, Alina and Cui, Shumo and Kurganov, Alexander}, year={2017}, pages={139–180} }
@article{chertock_dudzinski_kurganov_lukacova-medvid'ova_2018, title={Well-balanced schemes for the shallow water equations with Coriolis forces}, volume={138}, ISSN={["0945-3245"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1007/s00211-017-0928-0}, DOI={10.1007/s00211-017-0928-0}, number={4}, journal={NUMERISCHE MATHEMATIK}, author={Chertock, Alina and Dudzinski, Michael and Kurganov, Alexander and Lukacova-Medvid'ova, Maria}, year={2018}, month={Apr}, pages={939–973} }
@article{bernstein_chertock_kurganov_2016, title={Central-upwind scheme for shallow water equations with discontinuous bottom topography}, volume={47}, ISSN={["1678-7714"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1007/s00574-016-0124-3}, DOI={10.1007/s00574-016-0124-3}, number={1}, journal={BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY}, author={Bernstein, Andrew and Chertock, Alina and Kurganov, Alexander}, year={2016}, month={Mar}, pages={91–103} }
@article{chertock_cui_kurganov_wu_2015, title={Steady State and Sign Preserving Semi-Implicit Runge--Kutta Methods for ODEs with Stiff Damping Term}, volume={53}, ISSN={0036-1429 1095-7170}, url={http://dx.doi.org/10.1137/151005798}, DOI={10.1137/151005798}, abstractNote={In this paper, we develop a family of second-order semi-implicit time integration methods for systems of ordinary differential equations (ODEs) with stiff damping term. The important feature of the new methods resides in the fact that they are capable of exactly preserving the steady states as well as maintaining the sign of the computed solution under the time step restriction determined by the nonstiff part of the system only. The new semi-implicit methods are based on the modification of explicit strong stability preserving Runge--Kutta (SSP-RK) methods and are proven to have a formal second order of accuracy, $A(\alpha)$-stability, and stiff decay. We illustrate the performance of the proposed SSP-RK based semi-implicit methods on both a scalar ODE example and a system of ODEs arising from the semi-discretization of the shallow water equations with stiff friction term. The obtained numerical results clearly demonstrate that the ability of the introduced ODE solver to exactly preserve equilibria plays ...}, number={4}, journal={SIAM Journal on Numerical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Chertock, Alina and Cui, Shumo and Kurganov, Alexander and Wu, Tong}, year={2015}, month={Jan}, pages={2008–2029} }
@article{chertock_cui_kurganov_wu_2015, title={Well-balanced positivity preserving central-upwind scheme for the shallow water system with friction terms}, volume={78}, ISSN={["1097-0363"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1002/fld.4023}, DOI={10.1002/fld.4023}, abstractNote={SummaryShallow water models are widely used to describe and study free‐surface water flow. While in some practical applications the bottom friction does not have much influence on the solutions, there are still many applications, where the bottom friction is important. In particular, the friction terms will play a significant role when the depth of the water is very small. In this paper, we study shallow water equations with friction terms and develop a semi‐discrete second‐order central‐upwind scheme that is capable of exactly preserving physically relevant steady states and maintaining the positivity of the water depth. The presence of the friction terms increases the level of complexity in numerical simulations as the underlying semi‐discrete system becomes stiff when the water depth is small. We therefore implement an efficient semi‐implicit Runge‐Kutta time integration method that sustains the well‐balanced and sign preserving properties of the semi‐discrete scheme. We test the designed method on a number of one‐dimensional and two‐dimensional examples that demonstrate robustness and high resolution of the proposed numerical approach. The data in the last numerical example correspond to the laboratory experiments reported in [L. Cea, M. Garrido, and J. Puertas, Journal of Hydrology, 382 (2010), pp. 88–102], designed to mimic the rain water drainage in urban areas containing houses. Since the rain water depth is typically several orders of magnitude smaller than the height of the houses, we develop a special technique, which helps to achieve a remarkable agreement between the numerical and experimental results. Copyright © 2015 John Wiley & Sons, Ltd.}, number={6}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS}, author={Chertock, A. and Cui, S. and Kurganov, A. and Wu, T.}, year={2015}, month={Jun}, pages={355–383} }
@article{carrillo_chertock_huang_2015, title={A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure}, volume={17}, ISSN={["1991-7120"]}, url={https://doi-org.prox.lib.ncsu.edu/10.4208/cicp.160214.010814a}, DOI={10.4208/cicp.160214.010814a}, abstractNote={AbstractWe propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure. These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential. The proposed scheme is able to cope with non-smooth stationary states, different time scales including metastability, as well as concentrations and self-similar behavior induced by singular nonlocal kernels. We use the scheme to explore properties of these equations beyond their present theoretical knowledge.}, number={1}, journal={COMMUNICATIONS IN COMPUTATIONAL PHYSICS}, author={Carrillo, Jose A. and Chertock, Alina and Huang, Yanghong}, year={2015}, month={Jan}, pages={233–258} }
@article{chertock_herty_kurganov_2014, title={An Eulerian-Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs}, volume={59}, ISSN={["1573-2894"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1007/s10589-014-9655-y}, DOI={10.1007/s10589-014-9655-y}, number={3}, journal={COMPUTATIONAL OPTIMIZATION AND APPLICATIONS}, author={Chertock, Alina and Herty, Michael and Kurganov, Alexander}, year={2014}, month={Dec}, pages={689–724} }
@article{chertock_liu_pendleton_2015, title={Elastic collisions among peakon solutions for the Camassa-Holm equation}, volume={93}, ISSN={["1873-5460"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1016/j.apnum.2014.01.001}, DOI={10.1016/j.apnum.2014.01.001}, abstractNote={The purpose of this paper is to study the dynamics of the interaction among a special class of solutions of the one-dimensional Camassa–Holm equation. The equation yields soliton solutions whose identity is preserved through nonlinear interactions. These solutions are characterized by a discontinuity at the peak in the wave shape and are thus called peakon solutions. We apply a particle method to the Camassa–Holm equation and show that the nonlinear interaction among the peakon solutions resembles an elastic collision, i.e., the total energy and momentum of the system before the peakon interaction is equal to the total energy and momentum of the system after the collision. From this result, we provide several numerical illustrations which support the analytical study, as well as showcase the merits of using a particle method to simulate solutions to the Camassa–Holm equation under a wide class of initial data.}, journal={APPLIED NUMERICAL MATHEMATICS}, author={Chertock, Alina and Liu, Jian-Guo and Pendleton, Terrance}, year={2015}, month={Jul}, pages={30–46} }
@article{castro diaz_cheng_chertock_kurganov_2014, title={Solving Two-Mode Shallow Water Equations Using Finite Volume Methods}, volume={16}, ISSN={["1991-7120"]}, url={https://doi-org.prox.lib.ncsu.edu/10.4208/cicp.180513.230514a}, DOI={10.4208/cicp.180513.230514a}, abstractNote={AbstractIn this paper, we develop and study numerical methods for the two-mode shallow water equations recently proposed in [S. STECHMANN, A. MAJDA, and B. KHOUIDER, Theor. Comput. Fluid Dynamics, 22 (2008), pp. 407-432]. Designing a reliable numerical method for this system is a challenging task due to its conditional hyperbolicity and the presence of nonconservative terms. We present several numerical approaches—two operator splitting methods (based on either Roe-type upwind or central-upwind scheme), a central-upwind scheme and a path-conservative central-upwind scheme—and test their performance in a number of numerical experiments. The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method.}, number={5}, journal={COMMUNICATIONS IN COMPUTATIONAL PHYSICS}, author={Castro Diaz, Manuel Jesus and Cheng, Yuanzhen and Chertock, Alina and Kurganov, Alexander}, year={2014}, month={Nov}, pages={1323–1354} }
@article{chertock_kurganov_liu_2014, title={Central-upwind schemes for the system of shallow water equations with horizontal temperature gradients}, volume={127}, ISSN={["0945-3245"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1007/s00211-013-0597-6}, DOI={10.1007/s00211-013-0597-6}, number={4}, journal={NUMERISCHE MATHEMATIK}, author={Chertock, Alina and Kurganov, Alexander and Liu, Yu}, year={2014}, month={Aug}, pages={595–639} }
@article{chertock_kurganov_polizzi_timofeyev_2013, title={PEDESTRIAN FLOW MODELS WITH SLOWDOWN INTERACTIONS}, volume={24}, ISSN={0218-2025 1793-6314}, url={http://dx.doi.org/10.1142/s0218202513400083}, DOI={10.1142/s0218202513400083}, abstractNote={ In this paper, we introduce and study one-dimensional models for the behavior of pedestrians in a narrow street or corridor. We begin at the microscopic level by formulating a stochastic cellular automata model with explicit rules for pedestrians moving in two opposite directions. Coarse-grained mesoscopic and macroscopic analogs are derived leading to the coupled system of PDEs for the density of the pedestrian traffic. The obtained first-order system of conservation laws is only conditionally hyperbolic. We also derive higher-order nonlinear diffusive corrections resulting in a parabolic macroscopic PDE model. Numerical experiments comparing and contrasting the behavior of the microscopic stochastic model and the resulting coarse-grained PDEs for various parameter settings and initial conditions are performed. These numerical experiments demonstrate that the nonlinear diffusion is essential for reproducing the behavior of the stochastic system in the nonhyperbolic regime. }, number={02}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World Scientific Pub Co Pte Lt}, author={Chertock, Alina and Kurganov, Alexander and Polizzi, Anthony and Timofeyev, Ilya}, year={2013}, month={Dec}, pages={249–275} }
@article{chertock_kurganov_qu_wu_2013, title={Three-Layer Approximation of Two-Layer Shallow Water Equations}, volume={18}, ISSN={["1648-3510"]}, url={https://doi-org.prox.lib.ncsu.edu/10.3846/13926292.2013.869269}, DOI={10.3846/13926292.2013.869269}, abstractNote={Two-layer shallow water equations describe flows that consist of two layers of inviscid fluid of different (constant) densities flowing over bottom topography. Unlike the single-layer shallow water system, the two-layer one is only conditionally hyperbolic: the system loses its hyperbolicity because of the momentum exchange terms between the layers and as a result its solutions may develop instabilities. We study a three-layer approximation of the two-layer shallow water equations by introducing an intermediate layer of a small depth. We examine the hyperbolicity range of the three-layer model and demonstrate that while it still may lose hyperbolicity, the three-layer approximation may improve stability properties of the two-layer shallow water system.}, number={5}, journal={MATHEMATICAL MODELLING AND ANALYSIS}, author={Chertock, Alina and Kurganov, Alexander and Qu, Zhuolin and Wu, Tong}, year={2013}, pages={675–693} }
@article{chertock_liu_pendleton_2012, title={CONVERGENCE OF A PARTICLE METHOD AND GLOBAL WEAK SOLUTIONS OF A FAMILY OF EVOLUTIONARY PDES}, volume={50}, ISSN={["1095-7170"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1137/110831386}, DOI={10.1137/110831386}, abstractNote={The purpose of this paper is to provide global existence and uniqueness results for a family of fluid transport equations by establishing convergence results for the particle method applied to these equations. The considered family of PDEs is a collection of strongly nonlinear equations which yield traveling wave solutions and can be used to model a variety of flows in fluid dynamics. We apply a particle method to the studied evolutionary equations and provide a new self-contained method for proving its convergence. The latter is accomplished by using the concept of space-time bounded variation and the associated compactness properties. From this result, we prove the existence of a unique global weak solution in some special cases and obtain stronger regularity properties of the solution than previously established.}, number={1}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={Chertock, Alina and Liu, Jian-Guo and Pendleton, Terrance}, year={2012}, pages={1–21} }
@inbook{chertock_liu_pendleton_2012, title={Convergence Analysis of the Particle Method for the Camassa-Holm Equation}, volume={18}, ISBN={9789814417068 9789814417099}, ISSN={2010-2259}, url={http://dx.doi.org/10.1142/9789814417099_0033}, DOI={10.1142/9789814417099_0033}, abstractNote={The purpose of this paper is to establish a new method for proving the convergence of the particle method applied to the Camassa-Holm (CH) equation. The CH equation is a strongly nonlinear, bi-Hamiltonian, completely integrable model in the context of shallow water waves. The equation admits solutions that are nonlinear superpositions of traveling waves that have a discontinuity in the first derivative at their peaks and therefore are called peakons. This behavior admits several diverse scientific applications, but introduce difficult numerical challenges. To accurately capture these solutions, one may apply the particle method to the CH equation. Using the concept of space-time bounded variation, we show that the particle solution converges to a global weak solution of the CH equation for positive Radon measure initial data. ∗The work of A. Chertock and T. Pendleton was supported in part by the NSF Grant DMS-0712898; the work of J.-G. Liu was supported in part by the NSF Grant DMS 10-11738}, booktitle={Series in Contemporary Applied Mathematics}, publisher={Co-Published with Higher Education Press}, author={Chertock, Alina and Liu, Jian-Guo and Pendleton, Terrance}, year={2012}, month={Dec}, pages={365–373} }
@article{chertock_du toit_marsden_2012, title={INTEGRATION OF THE EPDIFF EQUATION BY PARTICLE METHODS}, volume={46}, ISSN={["0764-583X"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1051/m2an/2011054}, DOI={10.1051/m2an/2011054}, abstractNote={The purpose of this paper is to apply particle methods to the numerical solution of the EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. This behavior allows for the description of many rich physical applications, but also introduces difficult numerical challenges. We present a particle method for the EPDiff equation that is well-suited for this class of solutions and for simulating collisions between wavefronts. Discretization by means of the particle method is shown to preserve the basic Hamiltonian, the weak and variational structure of the original problem, and to respect the conservation laws associated with symmetry under the Euclidean group. Numerical results illustrate that the particle method has superior features in both one and two dimensions, and can also be effectively implemented when the initial data of interest lies on a submanifold.}, number={3}, journal={ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE}, author={Chertock, Alina and Du Toit, Philip and Marsden, Jerrold Eldon}, year={2012}, pages={515–534} }
@article{chertock_kurganov_wang_wu_2012, title={ON A CHEMOTAXIS MODEL WITH SATURATED CHEMOTACTIC FLUX}, volume={5}, ISSN={["1937-5093"]}, url={https://doi-org.prox.lib.ncsu.edu/10.3934/krm.2012.5.51}, DOI={10.3934/krm.2012.5.51}, abstractNote={We propose a PDE chemotaxis model, which can be viewed as a regularization of the Patlak-Keller-Segel (PKS)
system. Our modification is based on a fundamental physical property of the chemotactic flux function---its
boundedness. This means that the cell velocity is proportional to the magnitude of the chemoattractant gradient
only when the latter is small, while when the chemoattractant gradient tends to infinity the cell velocity
saturates. Unlike the original PKS system, the solutions of the modified model do not blow up in either finite
or infinite time in any number of spatial dimensions, thus making it possible to use bounded spiky steady states
to model cell aggregation. After obtaining local and global existence results, we use the local and global
bifurcation theories to show the existence of one-dimensional spiky steady states; we also study the stability
of bifurcating steady states. Finally, we numerically verify these analytical results, and then demonstrate that
solutions of the two-dimensional model with nonlinear saturated chemotactic flux function typically develop very
complicated spiky structures.}, number={1}, journal={KINETIC AND RELATED MODELS}, author={Chertock, Alina and Kurganov, Alexander and Wang, Xuefeng and Wu, Yaping}, year={2012}, month={Mar}, pages={51–95} }
@article{chertock_fellner_kurganov_lorz_markowich_2012, title={Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach}, volume={694}, ISSN={0022-1120 1469-7645}, url={http://dx.doi.org/10.1017/jfm.2011.534}, DOI={10.1017/jfm.2011.534}, abstractNote={AbstractAquatic bacteria like Bacillus subtilis are heavier than water yet they are able to swim up an oxygen gradient and concentrate in a layer below the water surface, which will undergo Rayleigh–Taylor-type instabilities for sufficiently high concentrations. In the literature, a simplified chemotaxis–fluid system has been proposed as a model for bio-convection in modestly diluted cell suspensions. It couples a convective chemotaxis system for the oxygen-consuming and oxytactic bacteria with the incompressible Navier–Stokes equations subject to a gravitational force proportional to the relative surplus of the cell density compared to the water density. In this paper, we derive a high-resolution vorticity-based hybrid finite-volume finite-difference scheme, which allows us to investigate the nonlinear dynamics of a two-dimensional chemotaxis–fluid system with boundary conditions matching an experiment of Hillesdon et al. (Bull. Math. Biol., vol. 57, 1995, pp. 299–344). We present selected numerical examples, which illustrate (i) the formation of sinking plumes, (ii) the possible merging of neighbouring plumes and (iii) the convergence towards numerically stable stationary plumes. The examples with stable stationary plumes show how the surface-directed oxytaxis continuously feeds cells into a high-concentration layer near the surface, from where the fluid flow (recurring upwards in the space between the plumes) transports the cells into the plumes, where then gravity makes the cells sink and constitutes the driving force in maintaining the fluid convection and, thus, in shaping the plumes into (numerically) stable stationary states. Our numerical method is fully capable of solving the coupled chemotaxis–fluid system and enabling a full exploration of its dynamics, which cannot be done in a linearised framework.}, journal={Journal of Fluid Mechanics}, publisher={Cambridge University Press (CUP)}, author={Chertock, A. and Fellner, K. and Kurganov, A. and Lorz, A. and Markowich, P. A.}, year={2012}, month={Feb}, pages={155–190} }
@inbook{chertock_christov_kurganov_2011, title={Central-upwind schemes for Boussinesq paradigm equations}, volume={115}, url={https://doi-org.prox.lib.ncsu.edu/10.1007/978-3-642-17770-5_20}, DOI={10.1007/978-3-642-17770-5_20}, abstractNote={We develop a new accurate and robust numerical method for the Boussinesq paradigm equation (BPE). To design the method we first introduce a change of variables, for which the BPE takes the form of a nonlinear wave equation with the global pressure, and rewrite the wave equation as a system of conservation laws with a global flux. We then apply a Godunov-type central-upwind scheme together with an efficient FFT-based elliptic solver to the resulting system. Making use of the new scheme, we investigate the propagation of one- and two-dimensional solitary waves of BPE and identify their solitonic behaviour.}, booktitle={Computational science and high performance computing IV}, publisher={Springer, Berlin}, author={Chertock, Alina and Christov, Christo I. and Kurganov, Alexander}, year={2011}, pages={267–281} }
@article{chertock_doering_kashdan_kurganov_2010, title={A Fast Explicit Operator Splitting Method for Passive Scalar Advection}, volume={45}, ISSN={["1573-7691"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1007/s10915-010-9381-2}, DOI={10.1007/s10915-010-9381-2}, number={1-3}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Chertock, Alina and Doering, Charles R. and Kashdan, Eugene and Kurganov, Alexander}, year={2010}, month={Oct}, pages={200–214} }
@inbook{chertock_kurganov_2010, place={Roma}, title={On splitting-based numerical methods for convection-diffusion equations}, volume={24}, booktitle={Numerical Methods for Balance Laws, Quaderni di Matematica, Aracne editrice S.r.l}, author={Chertock, A. and Kurganov, A.}, year={2010}, pages={303} }
@inbook{chertock_kurganov_2010, place={Caserta}, series={Quaderni di Matematica}, title={On splitting-based numerical methods for convection-diffusionequations}, volume={24}, booktitle={Numerical Methods for Balance Laws}, publisher={Dipartimento di matematica della Seconda Universitá di Napoli}, author={Chertock, A. and Kurganov, A.}, editor={Puppo, G. and Russo, G.Editors}, year={2010}, pages={303}, collection={Quaderni di Matematica} }
@article{chertock_kurganov_2009, title={Computing multivalued solutions of pressureless gas dynamics
by deterministic particle methods}, volume={5}, url={http://global-sci.org/intro/article_detail/cicp/7750.html}, number={2-4}, journal={Commun. Comput. Phys.}, author={Chertock, Alina and Kurganov, Alexander}, year={2009}, month={Feb}, pages={565–581} }
@inproceedings{chertock_kurganov_2009, title={Computing multivalued solutions of pressureless gas dynamics by deterministic particle methods}, volume={5}, number={2-4}, booktitle={Communications in Computational Physics}, author={Chertock, A. and Kurganov, A.}, year={2009}, pages={565–581} }
@inbook{chertock_kurganov_2009, title={On splitting-based numerical methods for convection-diffusion
equations}, volume={24}, booktitle={Numerical methods for balance laws}, publisher={Dept. Math., Seconda Univ. Napoli, Caserta}, author={Chertock, Alina and Kurganov, Alexander}, year={2009}, pages={303–343} }
@article{chertock_kurganov_2008, title={A SIMPLE EULERIAN FINITE-VOLUME METHOD FOR COMPRESSIBLE FLUIDS IN DOMAINS WITH MOVING BOUNDARIES}, volume={6}, ISSN={["1539-6746"]}, url={http://projecteuclid.org.prox.lib.ncsu.edu/euclid.cms/1222716944}, DOI={10.4310/CMS.2008.v6.n3.a1}, abstractNote={We introduce a simple new Eulerian method for treatment of moving boundaries
in compressible fluid computations. Our approach is based on the extension
of the interface tracking method recently introduced in the context of
multifluids. The fluid domain is placed in a rectangular computational
domain of a fixed size, which is divided into Cartesian cells. At every
discrete time level, there are three types of cells: internal, boundary, and
external ones. The numerical solution is evolved in internal cells only. The
numerical fluxes at the cells near the boundary are computed using the
technique from [A. Chertock, S. Karni and A. Kurganov, M2AN Math. Model.
Numer. Anal., to appear] combined with a solid wall ghost-cell
extrapolation and an interpolation in the phase space. The proposed
computational framework is general and may be used in conjunction with one’s
favorite finite-volume method. The robustness of the new approach is
illustrated on a number of one- and two-dimensional numerical examples.}, number={3}, journal={COMMUNICATIONS IN MATHEMATICAL SCIENCES}, author={Chertock, Alina and Kurganov, Alexander}, year={2008}, month={Sep}, pages={531–556} }
@article{chertock_kurganov_2008, title={A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models}, volume={111}, ISSN={["0945-3245"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1007/s00211-008-0188-0}, DOI={10.1007/s00211-008-0188-0}, number={2}, journal={NUMERISCHE MATHEMATIK}, author={Chertock, Alina and Kurganov, Alexander}, year={2008}, month={Dec}, pages={169–205} }
@article{chertock_karni_kurganov_2008, title={INTERFACE TRACKING METHOD FOR COMPRESSIBLE MULTIFLUIDS}, volume={42}, ISSN={["1290-3841"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1051/m2an:2008036}, DOI={10.1051/m2an:2008036}, abstractNote={This paper is concerned with numerical methods for compressible multicomponent fluids. The fluid components are assumed immiscible, and are separated by material interfaces, each endowed with its own equation of state (EOS). Cell averages of computational cells that are occupied by several fluid components require a "mixed-cell" EOS, which may not always be physically meaningful, and often leads to spurious oscillations. We present a new interface tracking algorithm, which avoids using mixed-cell information by solving the Riemann problem between its single-fluid neighboring cells. The resulting algorithm is oscillation-free for isolated material interfaces, conservative, and tends to produce almost perfect jumps across material fronts. The computational framework is general and may be used in conjunction with one's favorite finite-volume method. The robustness of the method is illustrated on shock-interface interaction in one space dimension, oscillating bubbles with radial symmetry and shock-bubble interaction in two space dimensions.}, number={6}, journal={ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE}, author={Chertock, Alina and Karni, Smadar and Kurganov, Alexander}, year={2008}, pages={991–1019} }
@article{chertock_gottlieb_solomonoff_2008, title={Modified Optimal Prediction and its Application to a Particle-Method Problem}, volume={37}, ISSN={["1573-7691"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1007/s10915-008-9242-4}, DOI={10.1007/s10915-008-9242-4}, number={2}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Chertock, Alina and Gottlieb, David and Solomonoff, Alex}, year={2008}, month={Nov}, pages={189–201} }
@inbook{chertock_kashdan_kurganov_benzonigavage_serre_2008, title={Propagation of Diffusing Pollutant by a Hybrid Eulerian–Lagrangian Method}, ISBN={9783540757115 9783540757122}, url={http://dx.doi.org/10.1007/978-3-540-75712-2_33}, DOI={10.1007/978-3-540-75712-2_33}, booktitle={Hyperbolic Problems: Theory, Numerics, Applications}, publisher={Springer Berlin Heidelberg}, author={Chertock, Alina and Kashdan, E. and Kurganov, A. and BenzoniGavage, S and Serre, D}, year={2008}, pages={371–379} }
@article{chertock_kurganov_rykov_2007, title={A new sticky particle method for pressureless gas dynamics}, volume={45}, ISSN={["1095-7170"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1137/050644124}, DOI={10.1137/050644124}, abstractNote={We first present a new sticky particle method for the system of pressureless gas dynamics. The method is based on the idea of sticky particles, which seems to work perfectly well for the models with point mass concentrations and strong singularity formations. In this method, the solution is sought in the form of a linear combination of $\delta$-functions, whose positions and coefficients represent locations, masses, and momenta of the particles, respectively. The locations of the particles are then evolved in time according to a system of ODEs, obtained from a weak formulation of the system of PDEs. The particle velocities are approximated in a special way using global conservative piecewise polynomial reconstruction technique over an auxiliary Cartesian mesh. This velocities correction procedure leads to a desired interaction between the particles and hence to clustering of particles at the singularities followed by the merger of the clustered particles into a new particle located at their center of mass. The proposed sticky particle method is then analytically studied. We show that our particle approximation satisfies the original system of pressureless gas dynamics in a weak sense, but only within a certain residual, which is rigorously estimated. We also explain why the relevant errors should diminish as the total number of particles increases. Finally, we numerically test our new sticky particle method on a variety of one- and two-dimensional problems as well as compare the obtained results with those computed by a high-resolution finite-volume scheme. Our simulations demonstrate the superiority of the results obtained by the sticky particle method that accurately tracks the evolution of developing discontinuities and does not smear the developing $\delta$-shocks.}, number={6}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={Chertock, Alina and Kurganov, Alexander and Rykov, Yurii}, year={2007}, pages={2408–2441} }
@article{chertock_kurganov_petrova_2009, title={Fast explicit operator splitting method for convection-diffusion equations}, volume={59}, ISSN={["1097-0363"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1002/fld.1355}, DOI={10.1002/fld.1355}, abstractNote={AbstractSystems of convection–diffusion equations model a variety of physical phenomena which often occur in real life. Computing the solutions of these systems, especially in the convection dominated case, is an important and challenging problem that requires development of fast, reliable and accurate numerical methods. In this paper, we propose a second‐order fast explicit operator splitting (FEOS) method based on the Strang splitting. The main idea of the method is to solve the parabolic problem via a discretization of the formula for the exact solution of the heat equation, which is realized using a conservative and accurate quadrature formula. The hyperbolic problem is solved by a second‐order finite‐volume Godunov‐type scheme.We provide a theoretical estimate for the convergence rate in the case of one‐dimensional systems of linear convection–diffusion equations with smooth initial data. Numerical convergence studies are performed for one‐dimensional nonlinear problems as well as for linear convection–diffusion equations with both smooth and nonsmooth initial data. We finally apply the FEOS method to the one‐ and two‐dimensional systems of convection–diffusion equations which model the polymer flooding process in enhanced oil recovery. Our results show that the FEOS method is capable to achieve a remarkable resolution and accuracy in a very efficient manner, that is, when only few splitting steps are performed. Copyright © 2006 John Wiley & Sons, Ltd.}, number={3}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS}, author={Chertock, Alina and Kurganov, Alexander and Petrova, Guergana}, year={2009}, month={Jan}, pages={309–332} }
@article{chertock_kurganov_petrova_2006, title={Finite-volume-particle methods for models of transport of pollutant in shallow water}, volume={27}, ISSN={["1573-7691"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1007/s10915-005-9060-x}, DOI={10.1007/s10915-005-9060-x}, number={1-3}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Chertock, Alina and Kurganov, Alexander and Petrova, Guergana}, year={2006}, month={Jun}, pages={189–199} }
@article{chertock_kurganov_2006, title={On a practical implementation of particle methods}, volume={56}, ISSN={["1873-5460"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1016/j.apnum.2006.03.024}, DOI={10.1016/j.apnum.2006.03.024}, abstractNote={This paper is devoted to a practical implementation of deterministic particle methods for solving transport equations with discontinuous coefficients and/or initial data, and related problems. In such methods, the solution is sought in the form of a linear combination of the delta-functions, whose positions and coefficients represent locations and weights of the particles, respectively. The locations and weights of the particles are then evolved in time according to a system of ODEs, obtained from the weak formulation of the transport PDEs. The major theoretical difficulty in solving the resulting system of ODEs is the lack of smoothness of its right-hand side. While the existence of a generalized solution is guaranteed by the theory of Filippov, the uniqueness can only be obtained via a proper regularization. Another difficulty one may encounter is related to an interpretation of the computed solution, whose point values are to be recovered from its particle distribution. We demonstrate that some of known recovering procedures, suitable for smooth functions, may fail to produce reasonable results in the nonsmooth case, and discuss several successful strategies which may be useful in practice. Different approaches are illustrated in a number of numerical examples, including one- and two-dimensional transport equations and the reactive Euler equations of gas dynamics.}, number={10-11}, journal={APPLIED NUMERICAL MATHEMATICS}, author={Chertock, Alina and Kurganov, Alexander}, year={2006}, pages={1418–1431} }
@inbook{chertock_kurganov_2005, title={Conservative locally moving mesh method for multifluid flows}, booktitle={Finite Volumes for Complex Applications IV}, publisher={ISTE, London}, author={Chertock, A. and Kurganov, A.}, year={2005}, pages={273–284} }
@inbook{chertock_kurganov_petrova_2005, title={Fast explicit operator splitting method. Application to the polymer system}, volume={IV}, booktitle={Finite Volumes for Complex Applications IV}, publisher={ISTE, London}, author={Chertock, A. and Kurganov, A. and Petrova, G.}, year={2005}, pages={63–72} }
@article{chertock_kurganov_2004, title={On a hybrid finite-volume-particle method}, volume={38}, ISSN={["1290-3841"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1051/m2an:2004051}, DOI={10.1051/m2an:2004051}, abstractNote={We present a hybrid finite-volume-particle numerical method for computing the transport of a passive pollutant by a flow. The flow is modeled by the one- and two-dimensional Saint-Venant system of shallow water equations and the pollutant propagation is described by a transport equation. This paper is an extension of our previous work (Chertock, Kurganov and Petrova, J. Sci. Comput. (to appear)), where the one-dimensional finite-volume-particle method has been proposed. The core idea behind the finite-volume-particle method is to use different schemes for the flow and pollution computations: the shallow water equations are numerically integrated using a finite-volume scheme, while the transport equation is solved by a particle method. This way the specific advantages of each scheme are utilized at the right place. A special attention is given to the recovery of the point values of the numerical solution from its particle distribution. The reconstruction is obtained using a dual equation for the pollutant concentration. This results in a significantly enhanced resolution of the computed solution and also makes it much easier to extend the finite-volume-particle method to the two-dimensional case. Mathematics Subject Classification. 34A36, 35L67, 35Q35, 65M99.}, number={6}, journal={ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE}, author={Chertock, A and Kurganov, A}, year={2004}, pages={1071–1091} }
@article{chertock_kurganov_rosenau_2005, title={On degenerate saturated-diffusion equations with convection}, volume={18}, ISSN={["1361-6544"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1088/0951-7715/18/2/009}, DOI={10.1088/0951-7715/18/2/009}, abstractNote={We study a class of degenerate parabolic convection–diffusion equations, endowed with a mechanism for saturation of the diffusion flux, which corrects the unphysical gradient-flux relations at high gradients. This paper extends our previous works on the effects of diffusion with saturation on convection and the impact of saturation on porous media-type diffusion, where it has been demonstrated that a nonlinear saturating diffusion is susceptible to a self-induced formation of discontinuities. In this work we demonstrate that nonlinear convection enhances the breakdown effect. We carry both analytical and numerical studies of the model equation, ut + f(u)x = [φ(u)Q(ux, u)]x, where Q is a bounded increasing function, φ(0) = 0 and φ(u) ∼ un, n > 0 for u ∼ 0. Depending on a choice of n, we obtain two distinctive processes. If 0 ≤ n ≤ 1, a discontinuity forms only when the upstream–downstream disparity exceeds a critical threshold, but if n > 1, all travelling waves are found to have a sharp discontinuous front. In fact, given a compact or a semi-compact initial datum, the front will not start to move until such a discontinuity forms.}, number={2}, journal={NONLINEARITY}, author={Chertock, A and Kurganov, A and Rosenau, P}, year={2005}, month={Mar}, pages={609–630} }
@article{chertock_levy_2005, title={On wavelet-based numerical homogenization}, volume={3}, ISSN={["1540-3459"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1137/030600783}, DOI={10.1137/030600783}, abstractNote={Recently, a wavelet-based method was introduced for the systematic derivation of subgrid scale models in the numerical solution of partial differential equations. Starting from a discretization of the multiscale differential operator, the discrete operator is represented in a wavelet space and projected onto a coarser subspace. The coarse (homogenized) operator is then replaced by a sparse approximation to increase the efficiency of the resulting algorithm.In this work we show how to improve the efficiency of this numerical homogenization method by choosing a different compact representation of the homogenized operator. In two dimensions our approach for obtaining a sparse representation is significantly simpler than the alternative sparse representations. $L^{\infty}$ error estimates are derived for a sample elliptic problem. An additional improvement we propose is a natural fine-scales correction that can be implemented in the final homogenization step. This modification of the scheme improves the resol...}, number={1}, journal={MULTISCALE MODELING & SIMULATION}, author={Chertock, A and Levy, D}, year={2005}, pages={65–88} }
@article{chertock_kurganov_rosenau_2003, title={Formation of discontinuities in flux-saturated degenerate parabolic equations}, volume={16}, ISSN={["0951-7715"]}, url={https://doi-org.prox.lib.ncsu.edu/10.1088/0951-7715/16/6/301}, DOI={10.1088/0951-7715/16/6/301}, abstractNote={We endow the nonlinear degenerate parabolic equation used to describe propagation of thermal waves in plasma or in a porous medium, with a mechanism for flux saturation intended to correct the nonphysical gradient-flux relations at high gradients. We study both analytically and numerically the resulting equation: ut = [unQ(g(u)x)]x, n>0, where Q is a bounded increasing function. This model reveals that for n>1 the motion of the front is controlled by the saturation mechanism and instead of the typical infinite gradients resulting from the linear flux-gradients relations, Q∼ux, we obtain a sharp, shock-like front, typically associated with nonlinear hyperbolic phenomena. We prove that if the initial support is compact, independently of the smoothness of the initial datum inside the support, a sharp front discontinuity forms in a finite time, and until then the front does not expand.}, number={6}, journal={NONLINEARITY}, author={Chertock, A and Kurganov, A and Rosenau, P}, year={2003}, month={Nov}, pages={1875–1898} }
@article{chertock_levy_2002, title={A Particle Method for the KdV Equation}, volume={17}, url={https://doi-org.prox.lib.ncsu.edu/10.1023/A:1015106210404}, DOI={10.1023/A:1015106210404}, number={1-4}, journal={Journal of Scientific Computing}, author={Chertock, Alina and Levy, Doron}, year={2002}, pages={491–499} }
@article{chertock_2002, title={On the stability of a class of self-similar solutions to the filtration-absorption equation}, volume={13}, ISSN={0956-7925 1469-4425}, url={http://dx.doi.org/10.1017/s095679250100482x}, DOI={10.1017/s095679250100482x}, abstractNote={We consider the one-dimensional and two-dimensional filtration-absorption equation ut
= uΔu−(c−1)(∇u)2. The one-dimensional
case was considered previously by Barenblatt et al. [4],
where a special class of self-similar solutions was introduced. By the analogy with the 1D
case we construct a family of axisymmetric solutions in 2D. We demonstrate numerically that
the self-similar solutions obtained attract the solutions of non-self-similar Cauchy problems
having the initial condition of compact support. The main analytical result we provide is the
linear stability of the above self-similar solutions both in the 1D case and in the 2D case.}, number={2}, journal={European Journal of Applied Mathematics}, publisher={Cambridge University Press (CUP)}, author={Chertock, Alina}, year={2002}, month={Apr}, pages={179–194} }
@article{chertock_levy_2002, title={Particle methods for the KdV equation}, volume={17}, journal={Journal of Scientific Computing}, author={Chertock, A. and Levy, D.}, year={2002}, pages={491–499} }
@article{chertock_levy_2001, title={Particle Methods for Dispersive Equations}, volume={171}, ISSN={0021-9991}, url={http://dx.doi.org/10.1006/jcph.2001.6803}, DOI={10.1006/jcph.2001.6803}, abstractNote={We introduce a new dispersion-velocity particle method for approximating solutions of linear and nonlinear dispersive equations. This is the first time in which particle methods are being used for solving such equations. Our method is based on an extension of the diffusion-velocity method of Degond and Mustieles (SIAM J. Sci. Stat. Comput.11(2), 293 (1990)) to the dispersive framework. The main analytical result we provide is the short time existence and uniqueness of a solution to the resulting dispersion-velocity transport equation. We numerically test our new method for a variety of linear and nonlinear problems. In particular we are interested in nonlinear equations which generate structures that have nonsmooth fronts. Our simulations show that this particle method is capable of capturing the nonlinear regime of a compacton–compacton type interaction.}, number={2}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Chertock, Alina and Levy, Doron}, year={2001}, month={Aug}, pages={708–730} }
@article{barenblatt_bertsch_chertock_prostokishin_2000, title={Self-similar intermediate asymptotics for a degenerate parabolic filtration-absorption equation}, volume={97}, ISSN={0027-8424 1091-6490}, url={http://dx.doi.org/10.1073/pnas.97.18.9844}, DOI={10.1073/pnas.97.18.9844}, abstractNote={
The equation ∂
t
u = u∂
xx
2
u −
(
c
− 1)(∂
x
u
)
2
is known in literature as a qualitative mathematical model of some biological phenomena. Here this equation is derived as a model of the groundwater flow in a water-absorbing fissurized porous rock; therefore, we refer to this equation as a filtration-absorption equation. A family of self-similar solutions to this equation is constructed. Numerical investigation of the evolution of non-self-similar solutions to the Cauchy problems having compactly supported initial conditions is performed. Numerical experiments indicate that the self-similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state: identical zero. An open problem caused by the nonuniqueness of the solution of the Cauchy problem is discussed.
}, number={18}, journal={Proceedings of the National Academy of Sciences}, publisher={Proceedings of the National Academy of Sciences}, author={Barenblatt, G. I. and Bertsch, M. and Chertock, A. E. and Prostokishin, V. M.}, year={2000}, month={Aug}, pages={9844–9848} }
@article{abarbanel_chertock_2000, title={Strict Stability of High-Order Compact Implicit Finite-Difference Schemes: The Role of Boundary Conditions for Hyperbolic PDEs, I}, volume={160}, ISSN={0021-9991}, url={http://dx.doi.org/10.1006/jcph.2000.6420}, DOI={10.1006/jcph.2000.6420}, abstractNote={Temporal, or “strict,” stability of approximation to PDEs is much more difficult to achieve than the “classical” Lax stability. In this paper, we present a class of finite-difference schemes for hyperbolic initial boundary value problems in one and two space dimensions that possess the property of strict stability. The approximations are constructed so that all eigenvalues of corresponding differentiation matrix have a nonpositive real part. Boundary conditions are imposed by using penalty-like terms. Fourth- and sixth-order compact implicit finite-difference schemes are constructed and analyzed. Computational efficacy of the approach is corroborated by a series of numerical tests in 1-D and 2-D scalar problems.}, number={1}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Abarbanel, Saul S. and Chertock, Alina E.}, year={2000}, month={May}, pages={42–66} }
@article{abarbanel_chertock_yefet_2000, title={Strict Stability of High-Order Compact Implicit Finite-Difference Schemes: The Role of Boundary Conditions for Hyperbolic PDEs, II}, volume={160}, ISSN={0021-9991}, url={http://dx.doi.org/10.1006/jcph.2000.6421}, DOI={10.1006/jcph.2000.6421}, abstractNote={Abstract This paper deals with the problem of systems of hyperbolic PDEs in one and two space dimensions, using the theory of part I [7].}, number={1}, journal={Journal of Computational Physics}, publisher={Elsevier BV}, author={Abarbanel, Saul S. and Chertock, Alina E. and Yefet, Amir}, year={2000}, month={May}, pages={67–87} }
@article{karamzin_trofimov_chertok_1991, title={An algorithm for the numerical solution of equations
describing processes in photoreceivers}, volume={3}, number={10}, journal={Mat. Model.}, author={Karamzin, Yu. N. and Trofimov, V. A. and Chertok, A. È.}, year={1991}, pages={95–103} }
@article{chertock_karamzin_trofimov_1991, title={On a numerical algorithm for nonlinear differential equations describing some processes in photoreceivers}, volume={3}, journal={Journal of Mathematical Modeling}, author={Chertock, A. and Karamzin, Y. and Trofimov, V.}, year={1991}, pages={95–103} }