A posteriori error estimates  for upwind finite volume schemes  for nonlinear conservation laws  in multi dimensions

Kröner, Dietmar; Ohlberger, Mario

doi:10.1090/s0025-5718-99-01158-8

Cited by 95 publications

(107 citation statements)

References 17 publications

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“…It was shown in certain model cases that E 1 (v h , t n ) is of higher order compared to E(v h , t n ). Thus the estimator E(v h , t n ) controls the error for a wide class of schemes (similar observations hold for finite volume schemes, for the monotone case see [19,30]). For systems of Conservation Laws there is no hope in general to prove such estimates.…”

Section: Adaptive Algorithmsmentioning

confidence: 82%

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Adaptive Finite Element Relaxation Schemes for Hyperbolic Conservation Laws

2001

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Abstract. We propose and study semidiscrete and fully discrete finite element schemes based on appropriate relaxation models for systems of Hyperbolic Conservation Laws. These schemes are using piecewise polynomials of arbitrary degree and their consistency error is of high order. The methods are combined with an adaptive strategy that yields fine mesh in shock regions and coarser mesh in the smooth parts of the solution. The computational performance of these methods is demonstrated by considering scalar problems and the system of elastodynamics.Mathematics Subject Classification. 35L65, 65M60, 65M50, 82C40.

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Section: Adaptive Algorithmsmentioning

confidence: 82%

“…This can be done with the help of a posteriori estimates. Such estimates are available, e.g., for finite difference schemes satisfying certain entropy inequalities [20] and for finite volume schemes [19,30], see also [11,12,26,39,48]. In particular consider first order schemes written in the (viscous) form…”

Section: Adaptive Algorithmsmentioning

confidence: 99%

Adaptive Finite Element Relaxation Schemes for Hyperbolic Conservation Laws

2001

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“…The theory behind the mesh adaptation technique for central schemes on unstructured staggered grids has been developed in [25,27]. We introduce the following three main steps of this technique.…”

Section: General Descriptionmentioning

confidence: 99%

“…For transient problems, following the theory of [25,27], for each edge e ij ∈ T h , we have the error estimate η eij :…”

Section: A Posteriori Error Estimatementioning

confidence: 99%

Three-Dimensional Adaptive Central Schemes on Unstructured Staggered Grids

Madrane¹

2008

Hyperbolic Problems: Theory, Numerics, Applications

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Abstract.We present an explicit second-order finite volume generalization of the onedimensional (1D) Nessyahu-Tadmor schemes for hyperbolic equations on adaptive unstructured tetrahedral grids. The nonoscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of the Riemann problem at the cell interfaces is bypassed thanks to the use of the staggered Lax-Friedrichs scheme, is extended here to a two-steps scheme. In order to reduce artificial viscosity, we start with an adaptively refined primal grid in three dimensions (3D), where the theoretical a posteriori result of the first-order scheme is used to derive appropriate refinement indicators. We apply those methods to solve Euler's equations. Numerical experimental tests on classical problems are obtained by our method and by the computational fluid dynamics software Fluent. These tests include results for the 3D Euler system (shock tube problem) and flow around an NACA0012 airfoil. 1. Introduction. The history of schemes on staggered grids can at least be traced back to the famous paper of Courant, Friedrichs, and Lewy in 1928 [13] in which they discovered a scheme on staggered grids for the linear wave equation in one-dimensional (1D). For a special system arising in fluid dynamic problems von Neumann and Richtmyer used staggered grids as well [38]. Four years later, Lax introduced the well-known Lax-Friedrichs scheme and analyzed it [29]. In 1990 Tadmor and Nessyahu [41] picked up the idea to use staggered grids, showed the connection to Godunov's method, and proposed a second-order extension to 1D systems.The main advantage of these schemes is that no information about solutions to local Riemann problems is needed. Using staggered grids one can replace the upwind fluxes by central differences. The price one has to pay is the occurrence of excessive numerical viscosity since the resulting scheme can be interpreted as a Lax-Friedrichs scheme. Therefore, a higher order scheme of monotone upstream-centered schemes for conservation laws (MUSCL)-type in one spatial dimension was proposed in [41]. Later in [4,3,6,7,8,36] central schemes were generalized to multidimensional schemes on unstructured grids. For a Cartesian grid, we refer to [9,22,30,28,31,39,40,45,32] for related work.In the case of staggered unstructured multidimensional grids, there exist only a few convergence results. In [5] convergence of a second-order central scheme on twodimensional (2D) grids has been proven for a linear conservation law. Convergence of the first-order Lax-Friedrichs scheme on the same staggered grids for nonlinear scalar problems has been proven in [19].

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“…E-mail: snicaise@univ-valenciennes.fr direction. See [12,21,1,10,11,18,19,30] for cell centered finite volume methods, [15,16,22] for vertex-centered methods, and [5,13,14] for finite volume element methods.…”

Section: Introductionmentioning

confidence: 99%

A Posteriori Error Estimates for Finite Volume Approximations

Cochez-Dhondt

Nicaise

Repin

2009

Math. Model. Nat. Phenom.

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Abstract. We present new a posteriori error estimates for the finite volume approximations of elliptic problems. They are obtained by applying functional a posteriori error estimates to natural extensions of the approximate solution and its flux computed by the finite volume method. The estimates give guaranteed upper bounds for the errors in terms of the primal (energy) norm, dual norm (for fluxes), and also in terms of the combined primal-dual norms. It is shown that the estimates provide sharp upper and lower bounds of the error and their practical computation requires solving only finite-dimensional problems.Key words: finite volume methods, elliptic problems, a posteriori error estimates of the functional type

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A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions

Cited by 95 publications

References 17 publications

Adaptive Finite Element Relaxation Schemes for Hyperbolic Conservation Laws

Adaptive Finite Element Relaxation Schemes for Hyperbolic Conservation Laws

Three-Dimensional Adaptive Central Schemes on Unstructured Staggered Grids

A Posteriori Error Estimates for Finite Volume Approximations

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