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60
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
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.
A thirdorder semidiscrete central scheme for conservation laws and convectiondiffusion equations
 SIAM J. SCI. COMPUT
, 2000
"... We present a new thirdorder, semidiscrete, central method for approximating solutions to multidimensional systems of hyperbolic conservation laws, convectiondiffusion equations, and related problems. Our method is a highorder extension of the recently proposed secondorder, semidiscrete method ..."
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Cited by 59 (5 self)
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We present a new thirdorder, semidiscrete, central method for approximating solutions to multidimensional systems of hyperbolic conservation laws, convectiondiffusion equations, and related problems. Our method is a highorder extension of the recently proposed secondorder, semidiscrete method in [16]. The method is derived independently of the specific piecewise polynomial reconstruction which is based on the previously computed cellaverages. We demonstrate our results, by focusing on the new thirdorder CWENO reconstruction presented in [21]. The numerical results we present, show the desired accuracy, high resolution and robustness of our method.
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.
HighResolution Nonoscillatory Central Schemes For HamiltonJacobi Equations
"... In this paper, we construct secondorder central schemes for multidimensional HamiltonJacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle. Thus, these schemes provide the first examples of nonoscillatory secondorder Godunovtype schemes based ..."
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Cited by 44 (4 self)
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In this paper, we construct secondorder central schemes for multidimensional HamiltonJacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle. Thus, these schemes provide the first examples of nonoscillatory secondorder Godunovtype schemes based on global projection operators. Numerical experiments are performed; L 1 /L # errors and convergence rates are calculated. For convex Hamiltonians, numerical evidence confirms that our central schemes converge with secondorder rates, when measured in the L 1 norm advocated in our recent paper [Numer Math, to appear]. The standard L # norm, however, fails to detect this secondorder rate.
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.
Central schemes on overlapping cells
, 2005
"... Nessyahu and Tadmor's central scheme [J. Comput. Phys. 87 (1990)] has the benefit of not using Riemann solvers for solving hyperbolic conservation laws. But the staggered averaging causes large dissipation when the time step size is small compared to the mesh size. The recent work of Kurganov ..."
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Cited by 17 (4 self)
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Nessyahu and Tadmor's central scheme [J. Comput. Phys. 87 (1990)] has the benefit of not using Riemann solvers for solving hyperbolic conservation laws. But the staggered averaging causes large dissipation when the time step size is small compared to the mesh size. The recent work of Kurganov and Tadmor [J. Comput. Phys. 160 (2000)] overcomes this problem by using a variable control volume and results in semidiscrete and fully discrete nonstaggered schemes. Motivated by this work, we introduce overlapping cell averages of the solution at the same discrete time level, and develop a simple alternative technique to control the O(1/Dt) dependence of the dissipation. The semidiscrete form of the central scheme can also be obtained to which the TVD Runge–Kutta time discretization methods of Shu and Osher [J. Comput. Phys. 77 (1988)] or other stable and sufficiently accurate ODE solvers can be applied. This technique is essentially independent of the reconstruction and the shape of the mesh. The overlapping cell representation of the solution also opens new possibilities for reconstructions. Generally speaking, more compact reconstruction can be achieved. In the following, schemes of up to fifth order in 1D and third order in 2D have been developed. We demonstrate through numerical examples that by combining two classes of the overlapping cells in the reconstruction we can achieve higher resolution.
Central discontinuous Galerkin methods on overlapping cells with a nonoscillatory hierarchical reconstruction
 SIAM J. Numer. Anal
, 2007
"... Abstract. The central scheme of Nessyahu and Tadmor [J. Comput. Phys, 87 (1990)] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of ..."
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Cited by 15 (7 self)
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Abstract. The central scheme of Nessyahu and Tadmor [J. Comput. Phys, 87 (1990)] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of Kurganov and Tadmor [J. Comput. Phys, 160 (2000)] employs a variable control volume, which in turn yields a semidiscrete nonstaggered central scheme. Another approach, which we advocate here, is to view the staggered meshes as a collection of overlapping cells and to realize the computed solution by its overlapping cell averages. This leads to a simple technique to avoid the excessive numerical dissipation for small time steps [Y. Liu; J. Comput. Phys, 209 (2005)]. At the heart of the proposed approach is the evolution of two pieces of information per cell, instead of one cell average which characterizes all central and upwind Godunovtype finite volume schemes. Overlapping cells lend themselves to the development of a centraltype discontinuous Galerkin (DG) method, following the series of work by Cockburn and Shu [J. Comput. Phys. 141 (1998)] and the references therein. In this paper we develop a central DG technique for hyperbolic conservation laws, where we take advantage of the redundant representation of the solution on overlapping cells. The use of redundant overlapping cells opens new possibilities, beyond those of Godunovtype schemes. In particular, the