Numerical convergence rate for a diffusive limit of hyperbolic systems: $p$-system with damping

Christophe Berthon; Marianne Bessemoulin-Chatard; Hélène Mathis

doi:10.5802/smai-jcm.10

Numerical convergence rate for a diffusive limit of hyperbolic systems:

p

-system with damping

Christophe Berthon ¹; Marianne Bessemoulin-Chatard ¹; Hélène Mathis ¹

¹ Université de Nantes - Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629 - 2 rue de la Houssinière, BP 92208 - 44322 Nantes, France

The SMAI Journal of computational mathematics, Volume 2 (2016), pp. 99-119.

Abstract

This paper deals with diffusive limit of the $p$ -system with damping and its approximation by an Asymptotic Preserving (AP) Finite Volume scheme. Provided the system is endowed with an entropy-entropy flux pair, we give the convergence rate of classical solutions of the $p$ -system with damping towards the smooth solutions of the porous media equation using a relative entropy method. Adopting a semi-discrete scheme, we establish that the convergence rate is preserved by the approximated solutions. Several numerical experiments illustrate the relevance of this result.

Published online: 2016-09-11

MR Zbl

DOI: 10.5802/smai-jcm.10

Classification: 65M08, 65M12
Keywords: Asymptotic Preserving scheme, numerical convergence rate, relative entropy

Author's affiliations:

Christophe Berthon ¹; Marianne Bessemoulin-Chatard ¹; Hélène Mathis ¹

¹ Université de Nantes - Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629 - 2 rue de la Houssinière, BP 92208 - 44322 Nantes, France

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     title = {Numerical convergence rate for a diffusive limit of hyperbolic systems: $p$-system with damping},
     journal = {The SMAI Journal of computational mathematics},
     pages = {99--119},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {2},
     year = {2016},
     doi = {10.5802/smai-jcm.10},
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     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.10/}
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Christophe Berthon; Marianne Bessemoulin-Chatard; Hélène Mathis. Numerical convergence rate for a diffusive limit of hyperbolic systems: $p$-system with damping. The SMAI Journal of computational mathematics, Volume 2 (2016), pp. 99-119. doi : 10.5802/smai-jcm.10. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.10/

References
Cited by

[1] C. Berthon; R. Turpault Asymptotic preserving HLL schemes, Numer. Methods Partial Differential Equations, Volume 27 (2011), pp. 1396-1422 http://onlinelibrary.wiley.com/doi/10.1002/num.20586/pdf | DOI | MR | Zbl

[2] S. Bianchini; B. Hanouzet; R. Natalini Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math., Volume 60 (2007) no. 11, pp. 1559-1622 | DOI | MR | Zbl

[3] F. Blachère; R. Turpault An admissibility and asymptotic-preserving scheme for systems of conservation laws with source term on 2D unstructured meshes, Journal of Computational Physics (2016) | DOI | MR | Zbl

[4] C. Buet; B. Després; E. Franck Design of asymptotic preserving finite volume schemes for the hyperbolic heat equation on unstructured meshes, Numerische Mathematik, Volume 122 (2012) no. 2, pp. 227-278 | DOI | MR | Zbl

[5] C. Buet; B. Després; E. Franck; T. Leroy Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes, Math. of Comp. (2016) | DOI | Zbl

[6] C. Cancès; H. Mathis; N. Seguin Relative entropy for the finite volume approximation of strong solutions to systems of conservation laws (Submitted)

[7] C. Christoforou; A. Tzavaras Relative entropy for hyperbolic-parabolic systems and application to the constitutive theory of thermoviscoelasticity, ArXiv e-prints (2016)

[8] C. M. Dafermos The second law of thermodynamics and stability, Arch. Ration. Mech. Anal., Volume 70 (1979) no. 2, pp. 167-179 | DOI | MR | Zbl

[9] C. M. Dafermos Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325, Springer-Verlag, Berlin, 2010 | MR | Zbl

[10] R. J. DiPerna Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J., Volume 28 (1979) no. 1, pp. 137-188 | DOI | MR

[11] L. Gosse; G. Toscani An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math. Acad. Sci. Paris, Volume 334 (2002) no. 4, pp. 337 -342 http://www.sciencedirect.com/science/article/pii/S1631073X02022574 | DOI | MR | Zbl

[12] L. Gosse; G. Toscani Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes, SIAM J. Numer. Anal., Volume 41 (2003) no. 2, p. 641-658 (electronic) | DOI | MR | Zbl

[13] A. Harten; P. D. Lax; B. van Leer On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., Volume 25 (1983) no. 1, pp. 35-61 | DOI | MR | Zbl

[14] L. Hsiao; T.-P. Liu Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys., Volume 143 (1992) no. 3, pp. 599-605 | DOI | MR | Zbl

[15] S. Jin Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., Volume 21 (1999) no. 2, p. 441-454 (electronic) | DOI | MR | Zbl

[16] S. Jin; L. Pareschi; G. Toscani Diffusive Relaxation Schemes for Multiscale Discrete-Velocity Kinetic Equations, SIAM J. Numer. Anal., Volume 35 (1998) no. 6, pp. 2405-2439 | DOI | MR | Zbl

[17] V. Jovanović; C. Rohde Error estimates for finite volume approximations of classical solutions for nonlinear systems of hyperbolic balance laws, SIAM J. Numer. Anal., Volume 43 (2006) no. 6, p. 2423-2449 (electronic) | DOI | MR | Zbl

[18] S. Kawashima Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A, Volume 106 (1987) no. 1-2, pp. 169-194 | DOI | MR | Zbl

[19] C. Lattanzio; A. E. Tzavaras Relative Entropy in Diffusive Relaxation, SIAM J. Math. Anal., Volume 45 (2013) no. 3, pp. 1563-1584 | DOI | MR | Zbl

[20] P.L. Lions; G. Toscani Diffusive limit for finite velocity Boltzmann kinetic models, Revista Matematica Iberoamericana, Volume 13 (1997) no. 3, pp. 473-514 | DOI | MR | Zbl

[21] P. Marcati; A.J. Milani; P. Secchi Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system, Manuscripta Math., Volume 60 (1988) no. 1, pp. 49-69 | DOI | MR

[22] M. Mei Best Asymptotic Profile for Hyperbolic $p$ -System with Damping, SIAM J. Math. Anal., Volume 42 (2010) no. 1, pp. 1-23 | DOI | MR | Zbl

[23] G. Naldi; L. Pareschi Numerical schemes for kinetic equations in diffusive regimes, Appl. Math. Lett., Volume 11 (1998) no. 2, pp. 29-35 http://www.sciencedirect.com/science/article/pii/S0893965998000068 | DOI | MR | Zbl

[24] G. Naldi; L. Pareschi Numerical Schemes for Hyperbolic Systems of Conservation Laws with Stiff Diffusive Relaxation, SIAM J. Numer. Anal., Volume 37 (2000) no. 4, pp. 1246-1270 | DOI | MR | Zbl

[25] K. Nishihara Convergence Rates to Nonlinear Diffusion Waves for Solutions of System of Hyperbolic Conservation Laws with Damping, J. Differential Equations, Volume 131 (1996) no. 2, pp. 171 -188 http://www.sciencedirect.com/science/article/pii/S002203969690159X | DOI | MR | Zbl

[26] K. Nishihara Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping, J. Differential Equations, Volume 137 (1997) no. 2, pp. 384-395 | DOI | MR | Zbl

[27] K. Nishihara; W. Wang; T. Yang $L_{p}$ -convergence rate to nonlinear diffusion waves for $p$ -system with damping, J. Differential Equations, Volume 161 (2000) no. 1, pp. 191-218 | DOI | MR | Zbl

[28] A. E. Tzavaras Relative entropy in hyperbolic relaxation, Commun. Math. Sci., Volume 3 (2005) no. 2, pp. 119-132 http://projecteuclid.org/getRecord?id=euclid.cms/1118778271 | DOI | MR | Zbl

[29] C. J. van Duyn; L. A. Peletier A class of similarity solutions of the nonlinear diffusion equation, Nonlinear Anal., Volume 1 (1976/77) no. 3, pp. 223-233 | DOI | MR | Zbl

[30] C. J. van Duyn; L. A. Peletier Asymptotic behaviour of solutions of a nonlinear diffusion equation, Arch. Ration. Mech. Anal., Volume 65 (1977) no. 4, pp. 363-377 | DOI | MR | Zbl

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