Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes

Buet, Christophe; Després, Bruno; Franck, Emmanuel; Leroy, Thomas

doi:10.1090/mcom/3131

Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes
HTML articles powered by AMS MathViewer

by Christophe Buet, Bruno Després, Emmanuel Franck and Thomas Leroy;

Math. Comp. 86 (2017), 1147-1202

DOI: https://doi.org/10.1090/mcom/3131

Published electronically: September 12, 2016

PDF | Request permission

Abstract:

We prove the uniform AP convergence on unstructured meshes in 2D of a generalization of the Gosse-Toscani 1D scheme for the hyperbolic heat equation. This scheme is also a nodal extension in 2D of the Jin-Levermore scheme for the 1D case. In 2D, the proof is performed using a new diffusion scheme.

References

D. Aregba, M. Briani, and R. Natalini, Time asymptotic high order schemes for dissipative BGK hyperbolic systems, arXiv:1207.6279
Christophe Berthon, Pierre Charrier, and Bruno Dubroca, An HLLC scheme to solve the $M_1$ model of radiative transfer in two space dimensions, J. Sci. Comput. 31 (2007), no. 3, 347–389. MR 2320554, DOI 10.1007/s10915-006-9108-6
Christophe Berthon, Philippe G. LeFloch, and Rodolphe Turpault, Late-time/stiff-relaxation asymptotic-preserving approximations of hyperbolic equations, Math. Comp. 82 (2013), no. 282, 831–860. MR 3008840, DOI 10.1090/S0025-5718-2012-02666-4
Christophe Berthon and Rodolphe Turpault, Asymptotic preserving HLL schemes, Numer. Methods Partial Differential Equations 27 (2011), no. 6, 1396–1422. MR 2838300, DOI 10.1002/num.20586
I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar. 7 (1956), 81–94 (English, with Russian summary). MR 79154, DOI 10.1007/BF02022967
Christophe Buet, Stéphane Cordier, Brigitte Lucquin-Desreux, and Simona Mancini, Diffusion limit of the Lorentz model: asymptotic preserving schemes, M2AN Math. Model. Numer. Anal. 36 (2002), no. 4, 631–655. MR 1932307, DOI 10.1051/m2an:2002028
Christophe Buet, Bruno Després, and Emmanuel Franck, Design of asymptotic preserving finite volume schemes for the hyperbolic heat equation on unstructured meshes, Numer. Math. 122 (2012), no. 2, 227–278. MR 2969268, DOI 10.1007/s00211-012-0457-9
Christophe Buet, Bruno Després, and Emmanuel Franck, An asymptotic preserving scheme with the maximum principle for the $M_1$ model on distorded meshes, C. R. Math. Acad. Sci. Paris 350 (2012), no. 11-12, 633–638 (English, with English and French summaries). MR 2956157, DOI 10.1016/j.crma.2012.07.002
Emmanuel Franck, Christophe Buet, and Bruno Després, Asymptotic preserving finite volumes discretization for non-linear moment model on unstructured meshes, Finite volumes for complex applications VI. Problems & perspectives. Volume 1, 2, Springer Proc. Math., vol. 4, Springer, Heidelberg, 2011, pp. 467–474. MR 2882323, DOI 10.1007/978-3-642-20671-9_{4}9
Nicolas Crouseilles and Mohammed Lemou, An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits, Kinet. Relat. Models 4 (2011), no. 2, 441–477. MR 2786393, DOI 10.3934/krm.2011.4.441
N. Crouseilles and M. Roger, A dynamic multi-scale model for transient radiative transfer calculations, Journal of Quantitative Spectroscopy and Radiative Transfer (2013), 110-121.
Bruno Despres, Lax theorem and finite volume schemes, Math. Comp. 73 (2004), no. 247, 1203–1234. MR 2047085, DOI 10.1090/S0025-5718-03-01618-1
Bruno Després, Weak consistency of the cell-centered Lagrangian GLACE scheme on general meshes in any dimension, Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 41-44, 2669–2679. MR 2728819, DOI 10.1016/j.cma.2010.05.010
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 10.1016/S1570-8659(00)07005-8
Francis Filbet and Amélie Rambaud, Analysis of an asymptotic preserving scheme for relaxation systems, ESAIM Math. Model. Numer. Anal. 47 (2013), no. 2, 609–633. MR 3021700, DOI 10.1051/m2an/2012042
François Golse, Shi Jin, and C. David Levermore, The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method, SIAM J. Numer. Anal. 36 (1999), no. 5, 1333–1369. MR 1706766, DOI 10.1137/S0036142997315986
Laurent Gosse, A two-dimensional version of the Godunov scheme for scalar balance laws, SIAM J. Numer. Anal. 52 (2014), no. 2, 626–652. MR 3179555, DOI 10.1137/130925906
Laurent Gosse and Giuseppe Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math. Acad. Sci. Paris 334 (2002), no. 4, 337–342 (English, with English and French summaries). MR 1891014, DOI 10.1016/S1631-073X(02)02257-4
L. Gosse, Transient radiative transfer in the grey case: Well-balanced and asymptotic-preserving schemes built on Cases’s elementary solutions, Journal of Quantitative Spectroscopy and Radiative Transfer 112 (2011), 1995–2012.
Laurent Gosse, A two-dimensional version of the Godunov scheme for scalar balance laws, SIAM J. Numer. Anal. 52 (2014), no. 2, 626–652. MR 3179555, DOI 10.1137/130925906
Shi Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput. 21 (1999), no. 2, 441–454. MR 1718639, DOI 10.1137/S1064827598334599
Shi Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Riv. Math. Univ. Parma (N.S.) 3 (2012), no. 2, 177–216. MR 2964096
Shi Jin and C. David Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys. 126 (1996), no. 2, 449–467. MR 1404381, DOI 10.1006/jcph.1996.0149
Shi Jin and David Levermore, The discrete-ordinate method in diffusive regimes, Transport Theory Statist. Phys. 20 (1991), no. 5-6, 413–439. MR 1139994, DOI 10.1080/00411459108203913
Shi Jin, Lorenzo Pareschi, and Giuseppe Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal. 38 (2000), no. 3, 913–936. MR 1781209, DOI 10.1137/S0036142998347978
Shi Jin, Lorenzo Pareschi, and Giuseppe Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM J. Numer. Anal. 35 (1998), no. 6, 2405–2439. MR 1655853, DOI 10.1137/S0036142997315962
Mohammed Lemou and Luc Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput. 31 (2008), no. 1, 334–368. MR 2460781, DOI 10.1137/07069479X
Jian-Guo Liu and Luc Mieussens, Analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit, SIAM J. Numer. Anal. 48 (2010), no. 4, 1474–1491. MR 2684343, DOI 10.1137/090772770
C. Mazeran, Sur la structure mathematique et l’approximation numerique de l’hydrodynamique lagrangienne bidimentionnelle, PhD thesis, University Bordeaux 1, 2007.
Giovanni Naldi and Lorenzo Pareschi, Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation, SIAM J. Numer. Anal. 37 (2000), no. 4, 1246–1270. MR 1756424, DOI 10.1137/S0036142997328810