Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients

Karlsen, Kenneth H.; Risebro, Nils Henrik

doi:10.1051/m2an:2001114

Cited by 59 publications

(60 citation statements)

References 49 publications

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“…On the other hand, Evje and Karlsen [25] show that explicit monotone finite difference schemes, which were introduced by Harten et al [32] and Crandall and Majda [18] for conservation laws, converge to BV entropy solutions for initial-value problems of strongly degenerate parabolic equations. These results are extended to several space dimensions in [33]. Further analyses of finite volume schemes for degenerate parabolic equations include the works by Eymard et al [26] and Michel and Vovelle [39].…”

Section: Introductionmentioning

confidence: 86%

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension

Bürger¹,

Ruiz²,

Schneider³

et al. 2008

ESAIM: M2AN

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Abstract. We present a fully adaptive multiresolution scheme for spatially one-dimensional quasilinear strongly degenerate parabolic equations with zero-flux and periodic boundary conditions. The numerical scheme is based on a finite volume discretization using the Engquist-Osher numerical flux and explicit time stepping. An adaptive multiresolution scheme based on cell averages is then used to speed up the CPU time and the memory requirements of the underlying finite volume scheme, whose first-order version is known to converge to an entropy solution of the problem. A particular feature of the method is the storage of the multiresolution representation of the solution in a graded tree, whose leaves are the non-uniform finite volumes on which the numerical divergence is eventually evaluated. Moreover using the L 1 contraction of the discrete time evolution operator we derive the optimal choice of the threshold in the adaptive multiresolution method. Numerical examples illustrate the computational efficiency together with the convergence properties.Mathematics Subject Classification. 35L65, 35R05, 65M06, 76T20.

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Section: Introductionmentioning

confidence: 86%

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension

Bürger¹,

Ruiz²,

Schneider³

et al. 2008

ESAIM: M2AN

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“…These results are extended to several space dimensions in [47]. The convergence of finite volume schemes for initial-boundary value problems is proved in [44,48].…”

Section: Multiresolution Schemesmentioning

confidence: 92%

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

Bürger

Ruiz

Schneider³

et al. 2007

J Eng Math

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Abstract. A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the approximation of Engquist-Osher for the flux and explicit time stepping. An adaptive multiresolution scheme with cell averages is then used to speed up CPU time and memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for sake of data compression and to facilitate navigation. Applications to traffic flow with driver reaction and a clarifier-thickener model illustrate the efficiency of this method.

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“…Discretization of the aforementioned hyperbolic, porous medium, convection-diffusion, and ellipticparabolic equations by finite volume methods is quite standard by now and often used in engeneering practice. We refer to [48,31,3,44,45,61,57,68,79,49,67,12,10,11,42] and references therein for different convergence results and numerical experiments. For related works on linear elliptic problems, see [2,1,57,41,23,58,50,51,53,52] and the discussion in Section 8.…”

Section: Introductionmentioning

confidence: 99%

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

Andreïanov¹,

Boyer²,

Hubert³

2006

Numerical Methods Partial

129

275

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Abstract. We consider a class of doubly nonlinear degenerate hyperbolicparabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès [41]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and "hyperbolic" L ∞ weak-⋆ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna, . . . ). Our results cover the case of non-Lipschitz nonlinearities.

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Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients

Cited by 59 publications

References 49 publications

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

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