A note on entropy inequalities and error estimates for higher-order accurate finite volume schemes on irregular families of grids

Author:
Sebastian Noelle

Journal:
Math. Comp. **65** (1996), 1155-1163

MSC (1991):
Primary 35L65, 65M12, 65M15, 65M50

DOI:
https://doi.org/10.1090/S0025-5718-96-00737-5

MathSciNet review:
1344618

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Abstract | References | Similar Articles | Additional Information

Abstract: Recently, Cockburn, Coquel and LeFloch proved convergence and error estimates for higher-order finite volume schemes. Their result is based on entropy inequalities which are derived under restrictive assumptions on either the flux function or the numerical fluxes. Moreover, they assume that the spatial grid satisfies a standard regularity assumption. Using instead entropy inequalities derived in previous work by Kröner, Noelle and Rokyta and a weaker condition on the grid, we can generalize and simplify the error estimates.

**1.**R. Beinert, D. Kröner,*Finite volume methods with local mesh alignment in 2D*, Notes Numer. Fluid Mech. 46 (1994), 38--53.**2.**B. Cockburn, F. Coquel, and P. LeFloch,*An error estimate for finite volume methods for multidimensional conservation laws*, Math. Comp. 63 (1994), 77--103. MR**95d:65078****3.**B. Cockburn, F. Coquel, and P. LeFloch,*Convergence of the finite volume method for multidimensional conservation laws*, SIAM J. Numer. Anal. 32 (1995), 687--705. CMP**95:13****4.**B. Cockburn and P. A. Gremaud,*Error estimates for finite element methods for scalar conservation laws*, To appear in SIAM J. Numer. Anal.**5.**F. Coquel and P. LeFloch,*Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach*, Math. Comp. 57 (1991), 169--210. MR**91m:65229****6.**D. Kröner, S. Noelle, and M. Rokyta,*Convergence of higher-order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions*, Numer. Math. 71 (1995), 527--560. CMP**96:02****7.**N. Kuznetsov,*Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation*, USSR Comp. Math. and Math. Phys. 16 (1976), 105--119.**8.**S. Noelle,*Convergence of higher-order finite volume schemes on irregular grids*, Adv. Comp. Math. 3 (1995), 197--218. MR**95:10****9.**S. Osher,*Riemann solvers, the entropy condition and difference approximations*, SIAM J. Numer. Anal. 21 (1984), 217--235. MR**86d:65119****10.**J. Vila,*Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws: I. Explicit monotone schemes*, Math. Modelling Numer. Analysis 28 (1994), 267--295. CMP**94:12**

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Additional Information

**Sebastian Noelle**

Email:
noelle@iam.uni-bonn.de

DOI:
https://doi.org/10.1090/S0025-5718-96-00737-5

Keywords:
Multidimensional conservation law,
finite volume method,
discrete entropy inequality,
error estimate,
irregular grids

Received by editor(s):
March 21, 1995

Additional Notes:
Partially supported by Deutsche Forschungsgemeinschaft, SFB 256.

Article copyright:
© Copyright 1996
American Mathematical Society