Error estimate for the approximation of nonlinear conservation laws on bounded domains by the finite volume method

Ohlberger, Mario; Vovelle, Julien

doi:10.1090/S0025-5718-05-01770-9

Error estimate for the approximation of nonlinear conservation laws on bounded domains by the finite volume method
HTML articles powered by AMS MathViewer

by Mario Ohlberger and Julien Vovelle PDF

Math. Comp. 75 (2006), 113-150 Request permission

Abstract:

In this paper we derive a priori and a posteriori error estimates for cell centered finite volume approximations of nonlinear conservation laws on polygonal bounded domains. Numerical experiments show the applicability of the a posteriori result for the derivation of local adaptive solution strategies.

References

S. Benharbit, A. Chalabi, and J.-P. Vila, Numerical viscosity and convergence of finite volume methods for conservation laws with boundary conditions, SIAM J. Numer. Anal. 32 (1995), no. 3, 775–796. MR 1335655, DOI 10.1137/0732036
C. Bardos, A. Y. le Roux, and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations 4 (1979), no. 9, 1017–1034. MR 542510, DOI 10.1080/03605307908820117
José Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal. 147 (1999), no. 4, 269–361. MR 1709116, DOI 10.1007/s002050050152
B. Cockburn, F. Coquel, and P. G. LeFloch, Convergence of the finite volume method for multidimensional conservation laws, SIAM J. Numer. Anal. 32 (1995), no. 3, 687–705. MR 1335651, DOI 10.1137/0732032
B. Cockburn, F. Coquel, and P. G. LeFloch, Convergence of the finite volume method for multidimensional conservation laws, SIAM J. Numer. Anal. 32 (1995), no. 3, 687–705. MR 1335651, DOI 10.1137/0732032
Bernardo Cockburn and Huiing Gau, A posteriori error estimates for general numerical methods for scalar conservation laws, Mat. Apl. Comput. 14 (1995), no. 1, 37–47 (English, with English and Portuguese summaries). MR 1339916
Claire Chainais-Hillairet, Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate, M2AN Math. Model. Numer. Anal. 33 (1999), no. 1, 129–156 (English, with English and French summaries). MR 1685749, DOI 10.1051/m2an:1999109
Claire Chainais-Hillairet and Emmanuel Grenier, Numerical boundary layers for hyperbolic systems in 1-D, M2AN Math. Model. Numer. Anal. 35 (2001), no. 1, 91–106. MR 1811982, DOI 10.1051/m2an:2001100
J. Droniou, C. Imbert, and J. Vovelle, An error estimate for the parabolic approximation of multidimensional scalar conservation laws with boundary conditions, Ann. Inst. H. Poincaré C Anal. Non Linéaire 21 (2004), no. 5, 689–714 (English, with English and French summaries). MR 2086755, DOI 10.1016/j.anihpc.2003.11.001
R. Eymard, T. Gallouët, M. Ghilani, and R. Herbin, Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes, IMA J. Numer. Anal. 18 (1998), no. 4, 563–594. MR 1681074, DOI 10.1093/imanum/18.4.563
Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 713–1020. MR 1804748, DOI 10.1086/phos.67.4.188705
Robert Eymard, Thierry Gallouët, and Jullien Vovelle, Limit boundary conditions for finite volume approximations of some physical problems, J. Comput. Appl. Math. 161 (2003), no. 2, 349–369. MR 2017019, DOI 10.1016/j.cam.2003.05.003
Laurent Gosse and Charalambos Makridakis, Two a posteriori error estimates for one-dimensional scalar conservation laws, SIAM J. Numer. Anal. 38 (2000), no. 3, 964–988. MR 1781211, DOI 10.1137/S0036142999350383
Ralf Hartmann and Paul Houston, Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws, SIAM J. Sci. Comput. 24 (2002), no. 3, 979–1004. MR 1950521, DOI 10.1137/S1064827501389084
P. Houston, J. A. Mackenzie, E. Süli, and G. Warnecke, A posteriori error analysis for numerical approximations of Friedrichs systems, Numer. Math. 82 (1999), no. 3, 433–470. MR 1692130, DOI 10.1007/s002110050426
C. Imbert and J. Vovelle, A kinetic formulation for multidimensional scalar conservation laws with boundary conditions and applications, SIAM J. Math. Anal. 36 (2004), no. 1, 214–232. MR 2083859, DOI 10.1137/S003614100342468X
Claes Johnson and Anders Szepessy, Adaptive finite element methods for conservation laws based on a posteriori error estimates, Comm. Pure Appl. Math. 48 (1995), no. 3, 199–234. MR 1322810, DOI 10.1002/cpa.3160480302
Smadar Karni, Alexander Kurganov, and Guergana Petrova, A smoothness indicator for adaptive algorithms for hyperbolic systems, J. Comput. Phys. 178 (2002), no. 2, 323–341. MR 1899180, DOI 10.1006/jcph.2002.7024
Dietmar Kröner, Sebastian Noelle, and Mirko Rokyta, Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions, Numer. Math. 71 (1995), no. 4, 527–560. MR 1355053, DOI 10.1007/s002110050156
Dietmar Kröner and Mario Ohlberger, A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions, Math. Comp. 69 (2000), no. 229, 25–39. MR 1659843, DOI 10.1090/S0025-5718-99-01158-8
Marc Küther and Mario Ohlberger, Adaptive second order central schemes on unstructured staggered grids, Hyperbolic problems: theory, numerics, applications, Springer, Berlin, 2003, pp. 675–684. MR 2053216
S. N. Kružkov, First order quasilinear equations with several independent variables. , Mat. Sb. (N.S.) 81 (123) (1970), 228–255 (Russian). MR 0267257
N. N. Kuznecov, 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 (1976), no. 6, 1489–1502, 1627 (Russian). MR 483509
J. Málek, J. Nečas, M. Rokyta, and M. Růžička, Weak and measure-valued solutions to evolutionary PDEs, Applied Mathematics and Mathematical Computation, vol. 13, Chapman & Hall, London, 1996. MR 1409366, DOI 10.1007/978-1-4899-6824-1
M. Ohlberger, A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations, Numer. Math. 87 (2001), no. 4, 737–761. MR 1815733, DOI 10.1007/PL00005431
Mario Ohlberger, A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations, M2AN Math. Model. Numer. Anal. 35 (2001), no. 2, 355–387. MR 1825703, DOI 10.1051/m2an:2001119
Felix Otto, Initial-boundary value problem for a scalar conservation law, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 8, 729–734 (English, with English and French summaries). MR 1387428
Denis Serre, Systèmes de lois de conservation. II, Fondations. [Foundations], Diderot Editeur, Paris, 1996 (French, with French summary). Structures géométriques, oscillation et problèmes mixtes. [Geometric structures, oscillation and mixed problems]. MR 1459989
Endre Süli and Paul Houston, Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity, The state of the art in numerical analysis (York, 1996) Inst. Math. Appl. Conf. Ser. New Ser., vol. 63, Oxford Univ. Press, New York, 1997, pp. 441–471. MR 1628356
A. Szepessy, Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 6, 749–782 (English, with French summary). MR 1135992, DOI 10.1051/m2an/1991250607491
Eitan Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal. 28 (1991), no. 4, 891–906. MR 1111445, DOI 10.1137/0728048
Tao Tang, Error estimates of approximate solutions for nonlinear scalar conservation laws, Hyperbolic problems: theory, numerics, applications, Vol. I, II (Magdeburg, 2000) Internat. Ser. Numer. Math., 140, vol. 141, Birkhäuser, Basel, 2001, pp. 873–882. MR 1871175
M. H. Vignal, Schemas Volumes Finis pour des equations elliptiques ou hyperboliques avec conditions aux limites, convergence et estimations d’erreur, Thesis, L’Ecole Normale Superieure, Lyon, 1997.
Julien Vovelle, Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains, Numer. Math. 90 (2002), no. 3, 563–596. MR 1884231, DOI 10.1007/s002110100307