@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_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_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} }
 @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_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} }
 @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} }
 @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} }