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60
SemiDiscrete CentralUpwind Schemes for Hyperbolic Conservation Laws and HamiltonJacobi Equations
 SIAM J. Sci. Comput
, 2000
"... We introduce new Godunovtype semidiscrete central schemes for hyperbolic systems of conservation laws and HamiltonJacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation, and can be viewed as a generalization of the schemes from [26, 24 ..."
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Cited by 116 (22 self)
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We introduce new Godunovtype semidiscrete central schemes for hyperbolic systems of conservation laws and HamiltonJacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation, and can be viewed as a generalization of the schemes from [26, 24, 25] and [27]. The main advantages of the proposed central schemes are the high resolution, due to the smaller amount of the numerical dissipation, and the simplicity. There are no Riemann solvers and characteristic decomposition involved, and this makes them a universal tool for a wide variety of applications. At the same time, the developed schemes have an upwind nature, since they respect the directions of wave propagation by measuring the onesided local speeds. This is the reason why we call them centralupwind schemes. The constructed schemes are applied to various problems, such as the Euler equations of gas dynamics, the HamiltonJacobi equations with convex and nonconvex Hamiltoni...
Central WENO Schemes for Hyperbolic Systems of Conservation Laws
 MATH. MODEL. NUMER. ANAL
, 2001
"... We present a family of highorder, essentially nonoscillatory, central schemes for approximating solutions of hyperbolic systems of conservation laws. These schemes are based on a new centered version of the Weighed Essentially NonOscillatory (WENO) reconstruction of pointvalues from cellaverages ..."
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Cited by 59 (13 self)
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We present a family of highorder, essentially nonoscillatory, central schemes for approximating solutions of hyperbolic systems of conservation laws. These schemes are based on a new centered version of the Weighed Essentially NonOscillatory (WENO) reconstruction of pointvalues from cellaverages, which is then followed by an accurate approximation of the fluxes via a natural continuous extension of RungeKutta solvers. We explicitly construct the third and fourthorder scheme and demonstrate their highresolution properties in several numerical tests.
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,
Central Schemes For Balance Laws Of Relaxation Type
 SIAM J. NUMER. ANAL
, 2000
"... Several models in mathematical physics are described by quasilinear hyperbolic systems with source term and in several cases the production term can become stiff. Here suitable central numerical schemes for such problems are developed and applications to the Broadwell model and extended thermodyna ..."
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Cited by 28 (7 self)
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Several models in mathematical physics are described by quasilinear hyperbolic systems with source term and in several cases the production term can become stiff. Here suitable central numerical schemes for such problems are developed and applications to the Broadwell model and extended thermodynamics are presented. The numerical methods are a generalization of the NessyahuTadmor scheme to the nonhomogeneous case byincluding the cell averages of the production terms in the discrete balance equations. A second order scheme uniformlyaccurate in the relaxation parameter is derived and its properties analyzed. Numerical tests confirm the accuracy and robustness of the scheme.
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 thirdorder semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems
 NUMER. MATH. (2001) 88: 683–729
, 2001
"... ..."
Highorder finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD
 Int. J. Comput. Fluid Dyn
"... In recent years, high order numerical methods have been widely used in computational fluid dynamics (CFD), to effectively resolve complex flow features using meshes which are reasonable for today’s computers. In this paper, we review and compare three types of high order methods being used in CFD, n ..."
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Cited by 21 (1 self)
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In recent years, high order numerical methods have been widely used in computational fluid dynamics (CFD), to effectively resolve complex flow features using meshes which are reasonable for today’s computers. In this paper, we review and compare three types of high order methods being used in CFD, namely the weighted essentially nonoscillatory (WENO) finite difference methods, the WENO finite volume methods, and the discontinuous Galerkin (DG) finite element methods. We summarize the main features of these methods, from a practical user’s point of view, indicate their applicability and relative strength, and show a few selected numerical examples to demonstrate their performance on illustrative model CFD problems.
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