Alexander Kurganov | Tulane University (original) (raw)

Papers by Alexander Kurganov

Research paper thumbnail of Interaction of tropical cyclone-like vortices with sea-surface temperature anomalies and topography in a simple shallow-water atmospheric model

Physics of Fluids, 2021

In this paper, we show how the thermal effects affect trajectories, intensity, and formation of s... more In this paper, we show how the thermal effects affect trajectories, intensity, and formation of secondary structures during the passages of strong tropical cyclone-like vortices over oceanic warm and cold pools as well as over an island-type topography. Our results are obtained using the moist-convective thermal rotating shallow-water atmospheric model recently developed in [A. Kurganov et al., "Moist-convective thermal rotating shallow-water model," Phys. Fluids 32, 066601 (2020)]. This model introduces thermodynamics of the moist air and moist convection in the standard rotating shallow-water models and allows to include in the latter atmosphere-ocean interactions in an elementary way.

Research paper thumbnail of A well-balanced central-upwind scheme for the thermal rotating shallow water equations

Journal of Computational Physics, 2020

We develop a well-balanced central-upwind scheme for rotating shallow water model with horizontal... more We develop a well-balanced central-upwind scheme for rotating shallow water model with horizontal temperature and/or density gradients-the thermal rotating shallow water (TRSW). The scheme is designed using the flux globalization approach: first, the source terms are incorporated into the fluxes, which results in a hyperbolic system with global fluxes; second, we apply the Riemann-problem-solver-free central-upwind scheme to the rewritten system. We ensure that the resulting method is well-balanced by switching off the numerical diffusion when the computed solution is near (at) thermo-geostrophic equilibria. The designed scheme is successfully tested on a series of numerical examples. Motivated by future applications to large-scale motions in the ocean and atmosphere, the model is considered on the tangent plane to a rotating planet both in mid-latitudes and at the Equator. The numerical scheme is shown to be capable of quite accurately maintaining the equilibrium states in the presence of nontrivial topography and rotation. Prior to numerical simulations, an analysis of the TRSW model based on the use of Lagrangian variables is presented, allowing one to obtain criteria of existence and uniqueness of the equilibrium state, of the wave-breaking and shock formation, and of instability development out of given initial conditions. The established criteria are confirmed in the conducted numerical experiments.

Research paper thumbnail of Finite-volume schemes for shallow-water equations

Acta Numerica, 2018

Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal... more Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal areas, and other situations in which the water depth is much smaller than the horizontal length scale of motion. The classical shallow-water equations, the Saint-Venant system, were originally proposed about 150 years ago and still are used in a variety of applications. For many practical purposes, it is extremely important to have an accurate, efficient and robust numerical solver for the Saint-Venant system and related models. As their solutions are typically non-smooth and even discontinuous, finite-volume schemes are among the most popular tools. In this paper, we review such schemes and focus on one of the simplest (yet highly accurate and robust) methods: central-upwind schemes. These schemes belong to the family of Godunov-type Riemann-problem-solver-free central schemes, but incorporate some upwinding information about the local speeds of propagation, which helps to reduce an ex...

Research paper thumbnail of Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics

Networks & Heterogeneous Media, 2009

We first develop non-oscillatory central schemes for a traffic flow model with Arrhenius look-ahe... more We first develop non-oscillatory central schemes for a traffic flow model with Arrhenius look-ahead dynamics, proposed in [A. Sopasakis and M.A. Katsoulakis, SIAM J. Appl. Math., 66 (2006), pp. 921-944]. This model takes into account interactions of every vehicle with other vehicles ahead ("look-ahead" rule) and can be written as a one-dimensional scalar conservation law with a global flux. The proposed schemes are extensions of the nonoscillatory central schemes, which belong to a class of Godunov-type projectionevolution methods. In this framework, a solution, computed at a certain time, is first approximated by a piecewise polynomial function, which is then evolved to the next time level according to the integral form of the conservation law. Most Godunov-type schemes are based on upwinding, which requires solving (generalized) Riemann problems. However, no (approximate) Riemann problem solver is available for conservation laws with global fluxes. Therefore, central schemes, which are Riemann-problem-solver-free, are especially attractive for the studied traffic flow model. Our numerical experiments demonstrate high resolution, stability, and robustness of the proposed methods, which are used to numerically investigate both dispersive and smoothing effects of the global flux. We also modify the model by Sopasakis and Katsoulakis by introducing a more realistic, linear interaction potential that takes into account the fact that a car's speed is affected more by nearby vehicles than distant (but still visible) ones. The central schemes are extended to the modified model. Our numerical studies clearly suggest that in the case of a good visibility, the new model yields solutions that seem to better correspond to reality.

Research paper thumbnail of Well-Balanced Central-Upwind Scheme for Compressible Two-Phase Flows

We design a new well-balanced central-upwind scheme for compressible two- phase ows. The new sche... more We design a new well-balanced central-upwind scheme for compressible two- phase ows. The new scheme is an extension of the semi-discrete central-upwind scheme proposed in5. The novelty of the presented method is in a special discretization of non- conservative product terms, which are exactly balanced with the numerical uxes when the method is applied to void waves. The new scheme is simpler than its predecessor and extends the applicability of central-upwind schemes to several important test problems that remained out of reach in5.

Research paper thumbnail of New High-Resolution Semi-discrete Central Schemes for Hamilton–Jacobi Equations

Journal of Computational Physics, 2000

We introduce a new high-resolution central scheme for multidimensional Hamilton-Jacobi equations.... more We introduce a new high-resolution central scheme for multidimensional Hamilton-Jacobi equations. The scheme retains the simplicity of the non-oscillatory central schemes developed by C.-T. Lin and E. Tadmor (in press, SIAM J. Sci. Comput.), yet it enjoys a smaller amount of numerical viscosity, independent of 1/ t. By letting t ↓ 0 we obtain a new second-order central scheme in the particularly simple semi-discrete form, along the lines of the new semi-discrete central schemes recently introduced by the authors in the context of hyperbolic conservation laws. Fully discrete versions are obtained with appropriate Runge-Kutta solvers. The smaller amount of dissipation enables efficient integration of convection-diffusion equations, where the accumulated error is independent of a small time step dictated by the CFL limitation. The scheme is non-oscillatory thanks to the use of nonlinear limiters. Here we advocate the use of such limiters on second discrete derivatives, which is shown to yield an improved high resolution when compared to the usual limitation of first derivatives. Numerical experiments demonstrate the remarkable resolution obtained by the proposed new central scheme.

Research paper thumbnail of Quasi-Lagrangian Acceleration of Eulerian Methods

We present a simple and efficient strategy for the acceleration of explicit Eulerian methods for ... more We present a simple and efficient strategy for the acceleration of explicit Eulerian methods for multidimensional hyperbolic systems of conservation laws. The strategy is based on the Galilean invariance of dynamic equations and optimization of the reference frame, in which the equations are numerically solved. The optimal reference frame moves (locally in time) with the average characteristic speed of the system, and, in this sense, the resulting method is quasi-Lagrangian. This leads to the acceleration of the numerical computations thanks to the optimal CFL condition and automatic adjustment of the computational domain to the evolving part of the solution. We show that our quasi-Lagrangian acceleration procedure may also reduce the numerical dissipation of the underlying Eulerian method. This leads to a significantly enhanced resolution, especially in the supersonic case. We demonstrate a great potential of the proposed method on a number of numerical examples.

Research paper thumbnail of An adaptive well-balanced positivity preserving central-upwind scheme on quadtree grids for shallow water equations

Computers & Fluids, 2020

We present an adaptive well-balanced positivity preserving central-upwind scheme on quadtree grid... more We present an adaptive well-balanced positivity preserving central-upwind scheme on quadtree grids for shallow water equations. The use of quadtree grids results in a robust, efficient and highly accurate numerical method. The quadtree model is developed based on the well-balanced positivity preserving central-upwind scheme proposed in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5 (2007), pp. 133-160]. The designed scheme is well-balanced in the sense that it is capable of exactly preserving "lake-at-rest" steady states. In order to achieve this as well as to preserve positivity of water depth, a continuous piecewise bilinear interpolation of the bottom topography function is utilized. This makes the proposed scheme capable of modelling flows over discontinuous bottom topography. Local gradients are examined to determine new seeding points in grid refinement for the next timestep. Numerical examples demonstrate the promising performance of the central-upwind quadtree scheme.

Research paper thumbnail of Three-dimensional shallow water system: A relaxation approach

Journal of Computational Physics, 2017

We study a three-dimensional shallow water system, which is obtained from the threedimensional Na... more We study a three-dimensional shallow water system, which is obtained from the threedimensional Navier-Stokes equations after Reynolds averaging and under the simplifying hydrostatic pressure assumption. Since the three-dimensional shallow water system is generically not hyperbolic, it cannot be numerically solved using hyperbolic shock capturing schemes. At the same time, existing simple finite-difference and finite-volume methods may fail in simulations of unsteady flows with sharp gradients, such as dam-break and flood flows. To overcome this limitation, we propose a novel numerical method, which is based on a relaxation approach utilized to "hyperbolize" the three-dimensional shallow water system. The extended relaxation system is hyperbolic and we develop a second-order semi-discrete central-upwind scheme for it. The proposed numerical method can preserve "lake at rest" steady states and positivity of water depth over irregular bottom topography. The accuracy, stability and robustness of the developed numerical method is verified on five numerical experiments.

Research paper thumbnail of Well-balanced central-upwind scheme for a fully coupled shallow water system modeling flows over erodible bed

Journal of Computational Physics, 2015

Intense sediment transport and rapid bed evolution are frequently observed under highlyenergetic ... more Intense sediment transport and rapid bed evolution are frequently observed under highlyenergetic flows, and bed erosion sometimes is of the same magnitude as the flow itself. Simultaneous simulation of multiple physical processes requires a fully coupled system to achieve an accurate hydraulic and morphodynamical prediction. In this paper, we develop a high-order well-balanced finite-volume method for a new fully coupled two-dimensional hyperbolic system consisting of the shallow water equations with friction terms coupled with the equations modeling the sediment transport and bed evolution. The nonequilibrium sediment transport equation is used to predict the sediment concentration variation. Since both bed-load, sediment entrainment and deposition have significant effects on the bed evolution, an Exner-based equation is adopted together with the Grass bed-load formula and sediment entrainment and deposition models to calculate the morphological process. The resulting 5 × 5 hyperbolic system of balance laws is numerically solved using a Godunov-type central-upwind scheme on a triangular grid. A computationally expensive process of finding all of the eigenvalues of the Jacobian matrices is avoided: The upper/lower bounds on the largest/smallest local speeds of propagation are estimated using the Lagrange theorem. A special discretization of the bed-slope term is proposed to guarantee the well-balanced property of the designed scheme. The proposed fully coupled model is verified on a number of numerical experiments.

Research paper thumbnail of Numerical dissipation switch for two-dimensional central-upwind schemes

Mathematical Modelling and Numerical Analysis, 2021

We propose a numerical dissipation switch, which helps to control the amount of numerical dissipa... more We propose a numerical dissipation switch, which helps to control the amount of numerical dissipation present in central-upwind schemes. Our main goal is to reduce the numerical dissipation without risking oscillations. This goal is achieved with the help of a more accurate estimate of the local propagation speeds in the parts of the computational domain, which are near contact discontinuities and shears. To this end, we introduce a switch parameter, which depends on the distributions of energy in the x- and y-directions. The resulting new central-upwind is tested on a number of numerical examples, which demonstrate the superiority of the proposed method over the original central-upwind scheme.

Research paper thumbnail of A Simple Finite-Volume Method on a Cartesian Mesh for Pedestrian Flows with Obstacles

Springer Proceedings in Mathematics & Statistics, 2017

We consider a two-dimensional pedestrian flow model with obstacles governed by scalar hyperbolic ... more We consider a two-dimensional pedestrian 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 secondorder 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.

Research paper thumbnail of Fast Explicit Operator Splitting Method . Application to the Polymer System

Computing solutions of convection-diffusion equations, e specially in the convection dominated ca... more Computing solutions of convection-diffusion equations, e specially in the convection dominated case, is an important and challenging problem tha t requires development of fast, reliable numerical methods. We propose a second-order fast explicit operator splitting (FEOS) method based on the Strang splitting. The main idea of the met hod is to solve the parabolic problem via a discretization of the formula for the exact sol uti n of the heat equation, which is realized using a conservative and accurate quadrature fo mula. The hyperbolic problem is solved by a second-order finite-volume Godunov-type scheme . Th FEOS method is applied to the oneand two-dimensional systems modeling two phase m ulticomponent flow in porous media. Our results demonstrate that the method achieves a re m rkable resolution and accuracy in a very efficient manner, that is, when only few splitting st ep are performed. RÉSUMÉ.Le calcul de solutions d’équations de type convection-diff usion est, specialement dans...

Research paper thumbnail of Hybrid Finite-Volume-Particle Method for Dusty Gas Flows

The SMAI journal of computational mathematics, 2017

Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques... more Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/

Research paper thumbnail of A New Approach for Designing Moving-Water Equilibria Preserving Schemes for the Shallow Water Equations

Journal of Scientific Computing, 2019

We construct a new second-order moving-water equilibria preserving central-upwind scheme for the ... more We construct a new second-order moving-water equilibria preserving central-upwind scheme for the one-dimensional Saint-Venant system of shallow water equations. The idea is based on a reformulation of the source terms as integral in the flux function. Reconstruction of the flux variable yields then a third order equation that can be solved exactly. This procedure does not require any further modification of existing schemes. Several numerical tests are performed to verify the ability of the proposed scheme to accurately capture small perturbations of steady states.

Research paper thumbnail of Adaptive moving mesh upwind scheme for the two-species chemotaxis model

Computers & Mathematics with Applications, 2019

Chemotaxis systems are used to model the propagation, aggregation and pattern formation of bacter... more 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 phenomenonrapid 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-Medvid'ová, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), pp. 131-152] and [Chertock et al., Adv. Comput. Math., 44 (2018), pp. 327-350], 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.

Research paper thumbnail of An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions

Kinetic & Related Models, 2019

In this paper, we study two-dimensional multiscale chemotaxis models based on a combination of th... more 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

Research paper thumbnail of High-order positivity-preserving hybrid finite-volume-finite-difference methods for chemotaxis systems

Advances in Computational Mathematics, 2017

Chemotaxis refers to mechanisms by which cellular motion occurs in response to an external stimul... more Chemotaxis refers to mechanisms by which cellular motion occurs in response to an external stimulus, usually a chemical one. Chemotaxis phenomenon plays an important role in bacteria/cell aggregation and pattern formation mechanisms, as well as in tumor growth. A common property of all chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in rapid growth of solutions in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. There is consequently a need for accurate and computationally efficient numerical methods for Communicated by: Carlos Garcia-Cervera

Research paper thumbnail of Moving-water equilibria preserving central-upwind schemes for the shallow water equations

Communications in Mathematical Sciences, 2016

We construct a new second-order moving-water equilibria preserving central-upwind scheme for the ... more We construct a new second-order moving-water equilibria preserving central-upwind scheme for the one-dimensional Saint-Venant system of shallow water equations. Special reconstruction procedure and source term discretization are the key components that guarantee the resulting scheme is capable of exactly preserving smooth moving-water steady-state solutions and a draining time-step technique ensures positivity of the water depth. Several numerical experiments are performed to verify the well-balanced and positivity preserving properties as well as the ability of the proposed scheme to accurately capture small perturbations of moving-water steady states. We also demonstrate the advantage and importance of utilizing the new method over its still-water equilibria preserving counterpart.

Research paper thumbnail of Steady State and Sign Preserving Semi-Implicit Runge--Kutta Methods for ODEs with Stiff Damping Term

SIAM Journal on Numerical Analysis, 2015

In this paper, we develop a family of second-order semi-implicit time integration methods for sys... more 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(α)-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 an important role in achieving high resolution when a coarse grid is used.

Research paper thumbnail of Interaction of tropical cyclone-like vortices with sea-surface temperature anomalies and topography in a simple shallow-water atmospheric model

Physics of Fluids, 2021

In this paper, we show how the thermal effects affect trajectories, intensity, and formation of s... more In this paper, we show how the thermal effects affect trajectories, intensity, and formation of secondary structures during the passages of strong tropical cyclone-like vortices over oceanic warm and cold pools as well as over an island-type topography. Our results are obtained using the moist-convective thermal rotating shallow-water atmospheric model recently developed in [A. Kurganov et al., "Moist-convective thermal rotating shallow-water model," Phys. Fluids 32, 066601 (2020)]. This model introduces thermodynamics of the moist air and moist convection in the standard rotating shallow-water models and allows to include in the latter atmosphere-ocean interactions in an elementary way.

Research paper thumbnail of A well-balanced central-upwind scheme for the thermal rotating shallow water equations

Journal of Computational Physics, 2020

We develop a well-balanced central-upwind scheme for rotating shallow water model with horizontal... more We develop a well-balanced central-upwind scheme for rotating shallow water model with horizontal temperature and/or density gradients-the thermal rotating shallow water (TRSW). The scheme is designed using the flux globalization approach: first, the source terms are incorporated into the fluxes, which results in a hyperbolic system with global fluxes; second, we apply the Riemann-problem-solver-free central-upwind scheme to the rewritten system. We ensure that the resulting method is well-balanced by switching off the numerical diffusion when the computed solution is near (at) thermo-geostrophic equilibria. The designed scheme is successfully tested on a series of numerical examples. Motivated by future applications to large-scale motions in the ocean and atmosphere, the model is considered on the tangent plane to a rotating planet both in mid-latitudes and at the Equator. The numerical scheme is shown to be capable of quite accurately maintaining the equilibrium states in the presence of nontrivial topography and rotation. Prior to numerical simulations, an analysis of the TRSW model based on the use of Lagrangian variables is presented, allowing one to obtain criteria of existence and uniqueness of the equilibrium state, of the wave-breaking and shock formation, and of instability development out of given initial conditions. The established criteria are confirmed in the conducted numerical experiments.

Research paper thumbnail of Finite-volume schemes for shallow-water equations

Acta Numerica, 2018

Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal... more Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal areas, and other situations in which the water depth is much smaller than the horizontal length scale of motion. The classical shallow-water equations, the Saint-Venant system, were originally proposed about 150 years ago and still are used in a variety of applications. For many practical purposes, it is extremely important to have an accurate, efficient and robust numerical solver for the Saint-Venant system and related models. As their solutions are typically non-smooth and even discontinuous, finite-volume schemes are among the most popular tools. In this paper, we review such schemes and focus on one of the simplest (yet highly accurate and robust) methods: central-upwind schemes. These schemes belong to the family of Godunov-type Riemann-problem-solver-free central schemes, but incorporate some upwinding information about the local speeds of propagation, which helps to reduce an ex...

Research paper thumbnail of Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics

Networks & Heterogeneous Media, 2009

We first develop non-oscillatory central schemes for a traffic flow model with Arrhenius look-ahe... more We first develop non-oscillatory central schemes for a traffic flow model with Arrhenius look-ahead dynamics, proposed in [A. Sopasakis and M.A. Katsoulakis, SIAM J. Appl. Math., 66 (2006), pp. 921-944]. This model takes into account interactions of every vehicle with other vehicles ahead ("look-ahead" rule) and can be written as a one-dimensional scalar conservation law with a global flux. The proposed schemes are extensions of the nonoscillatory central schemes, which belong to a class of Godunov-type projectionevolution methods. In this framework, a solution, computed at a certain time, is first approximated by a piecewise polynomial function, which is then evolved to the next time level according to the integral form of the conservation law. Most Godunov-type schemes are based on upwinding, which requires solving (generalized) Riemann problems. However, no (approximate) Riemann problem solver is available for conservation laws with global fluxes. Therefore, central schemes, which are Riemann-problem-solver-free, are especially attractive for the studied traffic flow model. Our numerical experiments demonstrate high resolution, stability, and robustness of the proposed methods, which are used to numerically investigate both dispersive and smoothing effects of the global flux. We also modify the model by Sopasakis and Katsoulakis by introducing a more realistic, linear interaction potential that takes into account the fact that a car's speed is affected more by nearby vehicles than distant (but still visible) ones. The central schemes are extended to the modified model. Our numerical studies clearly suggest that in the case of a good visibility, the new model yields solutions that seem to better correspond to reality.

Research paper thumbnail of Well-Balanced Central-Upwind Scheme for Compressible Two-Phase Flows

We design a new well-balanced central-upwind scheme for compressible two- phase ows. The new sche... more We design a new well-balanced central-upwind scheme for compressible two- phase ows. The new scheme is an extension of the semi-discrete central-upwind scheme proposed in5. The novelty of the presented method is in a special discretization of non- conservative product terms, which are exactly balanced with the numerical uxes when the method is applied to void waves. The new scheme is simpler than its predecessor and extends the applicability of central-upwind schemes to several important test problems that remained out of reach in5.

Research paper thumbnail of New High-Resolution Semi-discrete Central Schemes for Hamilton–Jacobi Equations

Journal of Computational Physics, 2000

We introduce a new high-resolution central scheme for multidimensional Hamilton-Jacobi equations.... more We introduce a new high-resolution central scheme for multidimensional Hamilton-Jacobi equations. The scheme retains the simplicity of the non-oscillatory central schemes developed by C.-T. Lin and E. Tadmor (in press, SIAM J. Sci. Comput.), yet it enjoys a smaller amount of numerical viscosity, independent of 1/ t. By letting t ↓ 0 we obtain a new second-order central scheme in the particularly simple semi-discrete form, along the lines of the new semi-discrete central schemes recently introduced by the authors in the context of hyperbolic conservation laws. Fully discrete versions are obtained with appropriate Runge-Kutta solvers. The smaller amount of dissipation enables efficient integration of convection-diffusion equations, where the accumulated error is independent of a small time step dictated by the CFL limitation. The scheme is non-oscillatory thanks to the use of nonlinear limiters. Here we advocate the use of such limiters on second discrete derivatives, which is shown to yield an improved high resolution when compared to the usual limitation of first derivatives. Numerical experiments demonstrate the remarkable resolution obtained by the proposed new central scheme.

Research paper thumbnail of Quasi-Lagrangian Acceleration of Eulerian Methods

We present a simple and efficient strategy for the acceleration of explicit Eulerian methods for ... more We present a simple and efficient strategy for the acceleration of explicit Eulerian methods for multidimensional hyperbolic systems of conservation laws. The strategy is based on the Galilean invariance of dynamic equations and optimization of the reference frame, in which the equations are numerically solved. The optimal reference frame moves (locally in time) with the average characteristic speed of the system, and, in this sense, the resulting method is quasi-Lagrangian. This leads to the acceleration of the numerical computations thanks to the optimal CFL condition and automatic adjustment of the computational domain to the evolving part of the solution. We show that our quasi-Lagrangian acceleration procedure may also reduce the numerical dissipation of the underlying Eulerian method. This leads to a significantly enhanced resolution, especially in the supersonic case. We demonstrate a great potential of the proposed method on a number of numerical examples.

Research paper thumbnail of An adaptive well-balanced positivity preserving central-upwind scheme on quadtree grids for shallow water equations

Computers & Fluids, 2020

We present an adaptive well-balanced positivity preserving central-upwind scheme on quadtree grid... more We present an adaptive well-balanced positivity preserving central-upwind scheme on quadtree grids for shallow water equations. The use of quadtree grids results in a robust, efficient and highly accurate numerical method. The quadtree model is developed based on the well-balanced positivity preserving central-upwind scheme proposed in [A. Kurganov and G. Petrova, Commun. Math. Sci., 5 (2007), pp. 133-160]. The designed scheme is well-balanced in the sense that it is capable of exactly preserving "lake-at-rest" steady states. In order to achieve this as well as to preserve positivity of water depth, a continuous piecewise bilinear interpolation of the bottom topography function is utilized. This makes the proposed scheme capable of modelling flows over discontinuous bottom topography. Local gradients are examined to determine new seeding points in grid refinement for the next timestep. Numerical examples demonstrate the promising performance of the central-upwind quadtree scheme.

Research paper thumbnail of Three-dimensional shallow water system: A relaxation approach

Journal of Computational Physics, 2017

We study a three-dimensional shallow water system, which is obtained from the threedimensional Na... more We study a three-dimensional shallow water system, which is obtained from the threedimensional Navier-Stokes equations after Reynolds averaging and under the simplifying hydrostatic pressure assumption. Since the three-dimensional shallow water system is generically not hyperbolic, it cannot be numerically solved using hyperbolic shock capturing schemes. At the same time, existing simple finite-difference and finite-volume methods may fail in simulations of unsteady flows with sharp gradients, such as dam-break and flood flows. To overcome this limitation, we propose a novel numerical method, which is based on a relaxation approach utilized to "hyperbolize" the three-dimensional shallow water system. The extended relaxation system is hyperbolic and we develop a second-order semi-discrete central-upwind scheme for it. The proposed numerical method can preserve "lake at rest" steady states and positivity of water depth over irregular bottom topography. The accuracy, stability and robustness of the developed numerical method is verified on five numerical experiments.

Research paper thumbnail of Well-balanced central-upwind scheme for a fully coupled shallow water system modeling flows over erodible bed

Journal of Computational Physics, 2015

Intense sediment transport and rapid bed evolution are frequently observed under highlyenergetic ... more Intense sediment transport and rapid bed evolution are frequently observed under highlyenergetic flows, and bed erosion sometimes is of the same magnitude as the flow itself. Simultaneous simulation of multiple physical processes requires a fully coupled system to achieve an accurate hydraulic and morphodynamical prediction. In this paper, we develop a high-order well-balanced finite-volume method for a new fully coupled two-dimensional hyperbolic system consisting of the shallow water equations with friction terms coupled with the equations modeling the sediment transport and bed evolution. The nonequilibrium sediment transport equation is used to predict the sediment concentration variation. Since both bed-load, sediment entrainment and deposition have significant effects on the bed evolution, an Exner-based equation is adopted together with the Grass bed-load formula and sediment entrainment and deposition models to calculate the morphological process. The resulting 5 × 5 hyperbolic system of balance laws is numerically solved using a Godunov-type central-upwind scheme on a triangular grid. A computationally expensive process of finding all of the eigenvalues of the Jacobian matrices is avoided: The upper/lower bounds on the largest/smallest local speeds of propagation are estimated using the Lagrange theorem. A special discretization of the bed-slope term is proposed to guarantee the well-balanced property of the designed scheme. The proposed fully coupled model is verified on a number of numerical experiments.

Research paper thumbnail of Numerical dissipation switch for two-dimensional central-upwind schemes

Mathematical Modelling and Numerical Analysis, 2021

We propose a numerical dissipation switch, which helps to control the amount of numerical dissipa... more We propose a numerical dissipation switch, which helps to control the amount of numerical dissipation present in central-upwind schemes. Our main goal is to reduce the numerical dissipation without risking oscillations. This goal is achieved with the help of a more accurate estimate of the local propagation speeds in the parts of the computational domain, which are near contact discontinuities and shears. To this end, we introduce a switch parameter, which depends on the distributions of energy in the x- and y-directions. The resulting new central-upwind is tested on a number of numerical examples, which demonstrate the superiority of the proposed method over the original central-upwind scheme.

Research paper thumbnail of A Simple Finite-Volume Method on a Cartesian Mesh for Pedestrian Flows with Obstacles

Springer Proceedings in Mathematics & Statistics, 2017

We consider a two-dimensional pedestrian flow model with obstacles governed by scalar hyperbolic ... more We consider a two-dimensional pedestrian 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 secondorder 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.

Research paper thumbnail of Fast Explicit Operator Splitting Method . Application to the Polymer System

Computing solutions of convection-diffusion equations, e specially in the convection dominated ca... more Computing solutions of convection-diffusion equations, e specially in the convection dominated case, is an important and challenging problem tha t requires development of fast, reliable numerical methods. We propose a second-order fast explicit operator splitting (FEOS) method based on the Strang splitting. The main idea of the met hod is to solve the parabolic problem via a discretization of the formula for the exact sol uti n of the heat equation, which is realized using a conservative and accurate quadrature fo mula. The hyperbolic problem is solved by a second-order finite-volume Godunov-type scheme . Th FEOS method is applied to the oneand two-dimensional systems modeling two phase m ulticomponent flow in porous media. Our results demonstrate that the method achieves a re m rkable resolution and accuracy in a very efficient manner, that is, when only few splitting st ep are performed. RÉSUMÉ.Le calcul de solutions d’équations de type convection-diff usion est, specialement dans...

Research paper thumbnail of Hybrid Finite-Volume-Particle Method for Dusty Gas Flows

The SMAI journal of computational mathematics, 2017

Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques... more Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/

Research paper thumbnail of A New Approach for Designing Moving-Water Equilibria Preserving Schemes for the Shallow Water Equations

Journal of Scientific Computing, 2019

We construct a new second-order moving-water equilibria preserving central-upwind scheme for the ... more We construct a new second-order moving-water equilibria preserving central-upwind scheme for the one-dimensional Saint-Venant system of shallow water equations. The idea is based on a reformulation of the source terms as integral in the flux function. Reconstruction of the flux variable yields then a third order equation that can be solved exactly. This procedure does not require any further modification of existing schemes. Several numerical tests are performed to verify the ability of the proposed scheme to accurately capture small perturbations of steady states.

Research paper thumbnail of Adaptive moving mesh upwind scheme for the two-species chemotaxis model

Computers & Mathematics with Applications, 2019

Chemotaxis systems are used to model the propagation, aggregation and pattern formation of bacter... more 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 phenomenonrapid 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-Medvid'ová, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), pp. 131-152] and [Chertock et al., Adv. Comput. Math., 44 (2018), pp. 327-350], 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.

Research paper thumbnail of An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions

Kinetic & Related Models, 2019

In this paper, we study two-dimensional multiscale chemotaxis models based on a combination of th... more 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

Research paper thumbnail of High-order positivity-preserving hybrid finite-volume-finite-difference methods for chemotaxis systems

Advances in Computational Mathematics, 2017

Chemotaxis refers to mechanisms by which cellular motion occurs in response to an external stimul... more Chemotaxis refers to mechanisms by which cellular motion occurs in response to an external stimulus, usually a chemical one. Chemotaxis phenomenon plays an important role in bacteria/cell aggregation and pattern formation mechanisms, as well as in tumor growth. A common property of all chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in rapid growth of solutions in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. There is consequently a need for accurate and computationally efficient numerical methods for Communicated by: Carlos Garcia-Cervera

Research paper thumbnail of Moving-water equilibria preserving central-upwind schemes for the shallow water equations

Communications in Mathematical Sciences, 2016

We construct a new second-order moving-water equilibria preserving central-upwind scheme for the ... more We construct a new second-order moving-water equilibria preserving central-upwind scheme for the one-dimensional Saint-Venant system of shallow water equations. Special reconstruction procedure and source term discretization are the key components that guarantee the resulting scheme is capable of exactly preserving smooth moving-water steady-state solutions and a draining time-step technique ensures positivity of the water depth. Several numerical experiments are performed to verify the well-balanced and positivity preserving properties as well as the ability of the proposed scheme to accurately capture small perturbations of moving-water steady states. We also demonstrate the advantage and importance of utilizing the new method over its still-water equilibria preserving counterpart.

Research paper thumbnail of Steady State and Sign Preserving Semi-Implicit Runge--Kutta Methods for ODEs with Stiff Damping Term

SIAM Journal on Numerical Analysis, 2015

In this paper, we develop a family of second-order semi-implicit time integration methods for sys... more 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(α)-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 an important role in achieving high resolution when a coarse grid is used.