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59
New HighResolution Central Schemes for Nonlinear Conservation Laws and ConvectionDiffusion Equations
 J. Comput. Phys
, 2000
"... this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order )).In particular, our new central schemes maintain their highresolution ..."
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Cited by 226 (22 self)
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this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order )).In particular, our new central schemes maintain their highresolution independent of O(1/#t ), and letting #t 0, they admit a particularly simple semidiscrete formulation
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...
Weighted Essentially NonOscillatory Schemes on Triangular Meshes
 J. Comput. Phys
, 1998
"... In this paper we construct high order weighted essentially nonoscillatory (WENO) schemes on two dimensional unstructured meshes (triangles) in the finite volume formulation. We present third order schemes using a combination of linear polynomials, and fourth order schemes using a combination of qua ..."
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Cited by 72 (12 self)
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In this paper we construct high order weighted essentially nonoscillatory (WENO) schemes on two dimensional unstructured meshes (triangles) in the finite volume formulation. We present third order schemes using a combination of linear polynomials, and fourth order schemes using a combination of quadratic polynomials. Numerical examples are shown to demonstrate the accuracies and robustness of the methods for shock calculations.
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.
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.
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.
A thirdorder semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems
 NUMER. MATH. (2001) 88: 683–729
, 2001
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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.