×

Error estimate for finite volume scheme. (English) Zbl 1116.35089

In this paper, the authors developed the finite element volume scheme for error estimation of linear advection problems. A number of theorems and lemmas are developed for theoretical foundation, but no numerical experiments are performed for illustration.

MSC:

35L65 Hyperbolic conservation laws
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv HAL

References:

[1] Bouche, D., Ghidaglia, J.-M., Pascal, F.: Error estimate and the geometric corrector for the upwind finite volume method applied to the linear advection equation, preprint (2005) · Zbl 1094.65089
[2] Brenier Y. (1984). Averaged multivalued solutions for scalar conservation laws. SIAM J. Numer. Anal. 21(6): 1013–1037 · Zbl 0565.65054 · doi:10.1137/0721063
[3] Cockburn B., Coquel F., LeFloch P. (1994). An error estimate for finite volume methods for multidimensional conservation laws. Math. Comp. 63(207): 77–103 · Zbl 0855.65103 · doi:10.1090/S0025-5718-1994-1240657-4
[4] Cockburn, B., Gremaud, P.-A., Yang, J.X.: A priori error estimates for numerical methods for scalar conservation laws. III. Multidimensional flux-splitting monotone schemes on non-Cartesian grids. SIAM J. Numer. Anal. 35(5), 1775–1803 (electronic) (1998) · Zbl 0909.65058
[5] Crandall M.G., Tartar L. (1980). Some relations between nonexpansive and order preserving mappings. Proc. Amer. Math. Soc. 78(3): 385–390 · Zbl 0449.47059 · doi:10.1090/S0002-9939-1980-0553381-X
[6] Després, B.: Lax theorem and finite volume schemes. Math. Comp. 73(247), 1203–1234 (electronic) (2004) · Zbl 1053.65073
[7] Després, B.: An explicit a priori estimate for a finite volume approximation of linear advection on non-Cartesian grids. SIAM J. Numer. Anal. 42(2), 484–504 (electronic) (2004) · Zbl 1127.65322
[8] Després, B.: Convergence of non-linear finite volume schemes for linear transport. In: Notes from the XIth Jacques-Louis Lions Hispano-French School on Numerical Simulation in Physics and Engineering (Spanish). pp. 219–239 (2004)
[9] Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods, Handbook of numerical analysis, vol. VII, North-Holland, Amsterdam, pp. 713–1020 (2000) · Zbl 0981.65095
[10] Federer H. (1969). Geometric measure theory Die Grundlehren der mathematischen Band 153. Springer, Berlin Heidelberg New York
[11] Johnson C., Pitkäranta J. (1986). An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math. Comp. 46(173): 1–26 · Zbl 0618.65105 · doi:10.1090/S0025-5718-1986-0815828-4
[12] Kuznetsov, N.N.: The accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation, Ž. Vyčisl. Mat. i Mat. Fiz. 16(6):1489–1502, 1627 (1976) · Zbl 0354.35021
[13] Ohlberger M., Vovelle J. (2006). Error estimate for the approximation of non-linear conservation laws on bounded domains by the finite volume method. Math. Comp. 75(253): 113–150 · Zbl 1082.65112 · doi:10.1090/S0025-5718-05-01770-9
[14] Şabac F. (1997). The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws. SIAM J. Numer. Anal. 34(6): 2306–2318 · Zbl 0992.65099 · doi:10.1137/S003614299529347X
[15] Tihonov A.N., Samarskiĭ A.A. (1962). Homogeneous difference schemes on irregular meshes. Ž. Vyčisl. Mat. i Mat. Fiz. 2: 812–832
[16] Tang T., Teng Z.H. (1995). The sharpness of Kuznetsov’s \(O(\sqrt{\Delta x}) L^ 1\) -error estimate for monotone difference schemes.Math. Comp. 64(210): 581–589 · Zbl 0845.65053
[17] Vila J.-P. (1994). Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicit monotone schemes, RAIRO Modél. Math. Anal. Numer. 28(3): 267–295 · Zbl 0823.65087
[18] Vila J.-P., Villedieu P. (2003). Convergence of an explicit finite volume scheme for first order symmetric systems. Numer. Math. 94(3): 573–602 · Zbl 1030.65110 · doi:10.1007/s00211-002-0396-y
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.