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116
Compact central WENO schemes for multidimensional conservation laws
 SIAM J. Sci. Comput
, 2000
"... We present new third and fifthorder Godunovtype central schemes for approximating solutions of the HamiltonJacobi (HJ) equation in an arbitrary number of space dimensions. These are the first central schemes for approximating solutions of the HJ equations with an order of accuracy that is greate ..."
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Cited by 60 (12 self)
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We present new third and fifthorder Godunovtype central schemes for approximating solutions of the HamiltonJacobi (HJ) equation in an arbitrary number of space dimensions. These are the first central schemes for approximating solutions of the HJ equations with an order of accuracy that is greater than two. In two space dimensions we present two versions for the thirdorder scheme: one scheme that is based on a genuinely twodimensional Central WENO reconstruction, and another scheme that is based on a simpler dimensionbydimension reconstruction. The simpler dimensionbydimension variant is then extended to a multidimensional fifthorder scheme. Our numerical examples in one, two and three space dimensions verify the expected order of accuracy of the schemes. Key words. HamiltonJacobi equations, central schemes, high order, WENO, CWENO.
CENTRALUPWIND SCHEMES FOR THE SAINTVENANT SYSTEM
, 2002
"... We present one and twodimensional centralupwind schemes for approximating solutions of the SaintVenant system with source terms due to bottom topography. The SaintVenant system has steadystate solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challen ..."
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Cited by 52 (6 self)
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We present one and twodimensional centralupwind schemes for approximating solutions of the SaintVenant system with source terms due to bottom topography. The SaintVenant system has steadystate solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preserve this delicate balance with numerical schemes. Small perturbations of these states are also very difficult to compute. Our approach is based on extending semidiscrete central schemes for systems of hyperbolic conservation laws to balance laws. Special attention is paid to the discretization of the source term such as to preserve stationary steadystate solutions. We also prove that the secondorder version of our schemes preserves the nonnegativity of the height of the water. This important feature allows one to compute solutions for problems that include dry areas.
Solution of twodimensional Riemann problems for gas dynamics without Riemann problem solvers
 Numer. Methods Partial Differential Equations
, 2002
"... We report here on our numerical study of the twodimensional Riemann problem for the compressible Euler equations. Compared with the relatively simple 1D configurations, the 2D case consists of a plethora of geometric wave patterns that pose a computational challenge for highresolution methods. T ..."
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Cited by 44 (4 self)
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We report here on our numerical study of the twodimensional Riemann problem for the compressible Euler equations. Compared with the relatively simple 1D configurations, the 2D case consists of a plethora of geometric wave patterns that pose a computational challenge for highresolution methods. The main feature in the present computations of these 2D waves is the use of the Riemannsolversfree central schemes presented by Kurganov et al. This family of central schemes avoids the intricate and timeconsuming computation of the eigensystem of the problem and hence offers a considerably simpler alternative to upwind methods. The numerical results illustrate that despite their simplicity, the central schemes are able to recover with comparable high resolution, the various features observed in the earlier,
On the reduction of numerical dissipation in centralupwind schemes
 COMMUN. COMPUT. PHYS
"... We study centralupwind schemes for systems of hyperbolic conservation laws, recently introduced in [A. Kurganov, S. Noelle and G. Petrova, SIAM J. Sci. Comput., 23 (2001), pp. 707–740]. Similarly to the staggered central schemes, these schemes are central Godunovtype projectionevolution methods t ..."
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Cited by 28 (9 self)
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We study centralupwind schemes for systems of hyperbolic conservation laws, recently introduced in [A. Kurganov, S. Noelle and G. Petrova, SIAM J. Sci. Comput., 23 (2001), pp. 707–740]. Similarly to the staggered central schemes, these schemes are central Godunovtype projectionevolution methods that enjoy the advantages of high resolution, simplicity, universality, and robustness. At the same time, the centralupwind framework allows one to decrease a relatively large amount of numerical dissipation present at the staggered central schemes. In this paper, we present a modification of the onedimensional fully and semidiscrete centralupwind schemes, in which the numerical dissipation is reduced even further. The goal is achieved by a more accurate projection of the evolved quantities onto the original grid. In the semidiscrete case, the reduction of dissipation procedure leads to a new, less dissipative numerical flux. We also extend the new semidiscrete scheme to the twodimensional case via the rigorous, genuinely multidimensional derivation. The new semidiscrete schemes are tested on a number of numerical examples, where one can observe an improved resolution, especially of the contact waves.
On the construction, comparison, and local characteristic decomposition for highorder central WENO schemes
 J. Comput. Phys
"... In this paper, we review and construct fifth and ninthorder central weighted essentially nonoscillatory (WENO) schemes based on a finite volume formulation, staggered mesh, and continuous extension of Runge–Kutta methods for solving nonlinear hyperbolic conservation law systems. Negative linear we ..."
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Cited by 27 (1 self)
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In this paper, we review and construct fifth and ninthorder central weighted essentially nonoscillatory (WENO) schemes based on a finite volume formulation, staggered mesh, and continuous extension of Runge–Kutta methods for solving nonlinear hyperbolic conservation law systems. Negative linear weights appear in such a formulation and they are treated using the technique recently introduced by Shi et al. (J. Comput. Phys. 175, 108 (2002)). We then perform numerical computations and comparisons with the finite difference WENO schemes of Jiang and Shu (J. Comput.
A SecondOrder WellBalanced Positivity Preserving CentralUpwind Scheme for the SaintVenant System
 Communications in Mathematical Sciences
"... Abstract. A family of Godunovtype centralupwind schemes for the SaintVenant system of shallow water equations has been first introduced in [A. Kurganov and D. Levy, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 397–425]. Depending on the reconstruction step, the secondorder versions of the sche ..."
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Cited by 26 (3 self)
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Abstract. A family of Godunovtype centralupwind schemes for the SaintVenant system of shallow water equations has been first introduced in [A. Kurganov and D. Levy, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 397–425]. Depending on the reconstruction step, the secondorder versions of the schemes there could be made either wellbalanced or positivity preserving, but fail to satisfy both properties simultaneously. Here, we introduce an improved secondorder centralupwind scheme which, unlike its forerunners, is capable to both preserve stationary steady states (lake at rest) and to guarantee the positivity of the computed fluid depth. Another novel property of the proposed scheme is its applicability to models with discontinuous bottom topography. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of one and twodimensional examples. Key words. Hyperbolic systems of conservation and balance laws, semidiscrete centralupwind schemes, SaintVenant system of shallow water equations. AMS subject classifications. 65M99, 35L65 1.
Shock capturing and front tracking methods for granular avalanches
 J. Comput. Phys
, 2002
"... Shock formations are observed in granular avalanches when supercritical flow merges into a region of subcritical flow. In this paper we employ a shockcapturing numerical scheme for the onedimensional Savage–Hutter theory of granular flow to describe this phenomenon. A Lagrangian moving mesh scheme ..."
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Cited by 25 (6 self)
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Shock formations are observed in granular avalanches when supercritical flow merges into a region of subcritical flow. In this paper we employ a shockcapturing numerical scheme for the onedimensional Savage–Hutter theory of granular flow to describe this phenomenon. A Lagrangian moving mesh scheme applied to the nonconservative form of the equations reproduces smooth solutions of these free boundary problems very well, but fails when shocks are formed. A nonoscillatory central (NOC) difference scheme with TVD limiter or WENO cell reconstruction for the conservative equations is therefore introduced. For the avalanche free boundary problems it must be combined with a fronttracking method, developed here, to properly describe the margin evolution. It is found that this NOC scheme combined with the fronttracking module reproduces both the shock wave and the smooth solution accurately. A piecewise quadratic WENO reconstruction improves the smoothness of the solution near local extrema. The schemes are checked against exact solutions for (1) an upward moving shock wave, (2) the motion of a parabolic cap down an inclined plane, and (3) the motion of a parabolic cap down a curved slope ending in a flat runout region, where a shock is formed as the avalanche comes to a halt. c ○ 2002 Elsevier Science Key Words: granular avalanche; shockcapturing; nonoscillatory central scheme; free moving boundary; fronttracking. 1.
Solving the Euler Equations on Graphics Processing Units
 Comp. Sci.  ICCS
"... Abstract. The paper describes how one can use commodity graphics cards (GPUs) as a highperformance parallel computer to simulate the dynamics of ideal gases in two and three spatial dimensions. The dynamics is described by the Euler equations, and numerical approximations are computed using stateo ..."
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Cited by 18 (2 self)
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Abstract. The paper describes how one can use commodity graphics cards (GPUs) as a highperformance parallel computer to simulate the dynamics of ideal gases in two and three spatial dimensions. The dynamics is described by the Euler equations, and numerical approximations are computed using stateoftheart highresolution finitevolume schemes. These schemes are based upon an explicit time discretisation and are therefore ideal candidates for parallel implementation. 1
New Interior Penalty Discontinuous Galerkin Methods for the KellerSegel Chemotaxis Model
 SIAM J. NUMER. ANAL
, 2008
"... We develop a family of new interior penalty discontinuous Galerkin methods for the KellerSegel chemotaxis model. This model is described by a system of two nonlinear PDEs: a convectiondiffusion equation for the cell density coupled with a reactiondiffusion equation for the chemoattractant concent ..."
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Cited by 17 (3 self)
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We develop a family of new interior penalty discontinuous Galerkin methods for the KellerSegel chemotaxis model. This model is described by a system of two nonlinear PDEs: a convectiondiffusion equation for the cell density coupled with a reactiondiffusion equation for the chemoattractant concentration. It has been recently shown that the convective part of this system is of a mixed hyperbolicelliptic type, which may cause severe instabilities when the studied system is solved by straightforward numerical methods. Therefore, the first step in the derivation of our new methods is made by introducing the new variable for the gradient of the chemoattractant concentration and by reformulating the original KellerSegel model in the form of a convectiondiffusionreaction system with a hyperbolic convective part. We then design interior penalty discontinuous Galerkin methods for the rewritten KellerSegel system. Our methods employ the centralupwind numerical fluxes, originally developed in the context of finitevolume methods for hyperbolic systems of conservation laws. In this paper, we consider Cartesian grids and prove error estimates for the proposed highorder discontinuous Galerkin methods. Our proof is valid for preblowup times since we assume boundedness of the exact solution. We also show that the blowup time of the exact solution is bounded from above by the blowup time of our numerical solution. In the numerical tests presented below, we demonstrate that the obtained numerical solutions have no negative values and are oscillationfree, even though no slope limiting technique has been implemented.
HighOrder SemiDiscrete CentralUpwind Schemes for MultiDimensional HamiltonJacobi Equations
 IMA J. Numer. Anal
"... We present the first fifthorder, semidiscrete centralupwind method for approximating solutions of multidimensional HamiltonJacobi equations. Unlike most of the commonly used highorder upwind schemes, our scheme is formulated as a Godunovtype scheme. The scheme is based on the fluxes of Kur ..."
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Cited by 16 (5 self)
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We present the first fifthorder, semidiscrete centralupwind method for approximating solutions of multidimensional HamiltonJacobi equations. Unlike most of the commonly used highorder upwind schemes, our scheme is formulated as a Godunovtype scheme. The scheme is based on the fluxes of KurganovTadmor and KurganovNoellePetrova, and is derived for an arbitrary number of space dimensions. A theorem establishing the monotonicity of these fluxes is provided.