A note on entropy inequalities and error estimates for higherorder accurate finite volume schemes on irregular families of grids
Author:
Sebastian Noelle
Journal:
Math. Comp. 65 (1996), 11551163
MSC (1991):
Primary 35L65, 65M12, 65M15, 65M50
MathSciNet review:
1344618
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Abstract: Recently, Cockburn, Coquel and LeFloch proved convergence and error estimates for higherorder 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), 3853.
 2.
Bernardo
Cockburn, Frédéric
Coquel, and Philippe
LeFloch, An error estimate for finite volume
methods for multidimensional conservation laws, Math. Comp. 63 (1994), no. 207, 77–103. MR 1240657
(95d:65078), http://dx.doi.org/10.1090/S00255718199412406574
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B. Cockburn, F. Coquel, and P. LeFloch, Convergence of the finite volume method for multidimensional conservation laws, SIAM J. Numer. Anal. 32 (1995), 687705. CMP 95:13
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B. Cockburn and P. A. Gremaud, Error estimates for finite element methods for scalar conservation laws, To appear in SIAM J. Numer. Anal.
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Frédéric
Coquel and Philippe
LeFloch, Convergence of finite difference
schemes for conservation laws in several space dimensions: the corrected
antidiffusive flux approach, Math. Comp.
57 (1991), no. 195, 169–210. MR 1079010
(91m:65229), http://dx.doi.org/10.1090/S00255718199110790102
 6.
D. Kröner, S. Noelle, and M. Rokyta, Convergence of higherorder upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions, Numer. Math. 71 (1995), 527560. CMP 96:02
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N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a firstorder quasilinear equation, USSR Comp. Math. and Math. Phys. 16 (1976), 105119.
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S. Noelle, Convergence of higherorder finite volume schemes on irregular grids, Adv. Comp. Math. 3 (1995), 197218. MR 95:10
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Osher, Riemann solvers, the entropy condition, and difference
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no. 2, 217–235. MR 736327
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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), 267295. CMP 94:12
 1.
 R. Beinert, D. Kröner, Finite volume methods with local mesh alignment in 2D, Notes Numer. Fluid Mech. 46 (1994), 3853.
 2.
 B. Cockburn, F. Coquel, and P. LeFloch, An error estimate for finite volume methods for multidimensional conservation laws, Math. Comp. 63 (1994), 77103. 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), 687705. 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), 169210. MR 91m:65229
 6.
 D. Kröner, S. Noelle, and M. Rokyta, Convergence of higherorder upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions, Numer. Math. 71 (1995), 527560. CMP 96:02
 7.
 N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a firstorder quasilinear equation, USSR Comp. Math. and Math. Phys. 16 (1976), 105119.
 8.
 S. Noelle, Convergence of higherorder finite volume schemes on irregular grids, Adv. Comp. Math. 3 (1995), 197218. MR 95:10
 9.
 S. Osher, Riemann solvers, the entropy condition and difference approximations, SIAM J. Numer. Anal. 21 (1984), 217235. 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), 267295. CMP 94:12
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Additional Information
Sebastian Noelle
Email:
noelle@iam.unibonn.de
DOI:
http://dx.doi.org/10.1090/S0025571896007375
PII:
S 00255718(96)007375
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
