Comptes Rendus
no. 3
An analysis of over-relaxation in a kinetic approximation of systems of conservation laws
Florence Drui1 ; Emmanuel Franck1; Philippe Helluy1; Laurent Navoret1
1 IRMA, Université de Strasbourg & Inria Tonus, France
Comptes Rendus. Mécanique, Volume 347 (2019) no. 3, pp. 259-269.

The over-relaxation approach is an alternative to the Jin–Xin relaxation method in order to apply the equilibrium source term in a more precise way. This is also a key ingredient of the lattice Boltzmann method for achieving second-order accuracy. In this work, we provide an analysis of the over-relaxation kinetic scheme. We compute its equivalent equation, which is particularly useful for devising stable boundary conditions for the hidden kinetic variables.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2018.12.001
Keywords: Kinetic relaxation, Equivalent equation, Boundary conditions, Asymptotic preserving

Florence Drui 1; Emmanuel Franck 1; Philippe Helluy 1; Laurent Navoret 1

1 IRMA, Université de Strasbourg & Inria Tonus, France 
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Florence Drui; Emmanuel Franck; Philippe Helluy; Laurent Navoret. An analysis of over-relaxation in a kinetic approximation of systems of conservation laws. Comptes Rendus. Mécanique, Volume 347 (2019) no. 3, pp. 259-269. doi : 10.1016/j.crme.2018.12.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2018.12.001/

[1] S. Jin; Z. Xin The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Commun. Pure Appl. Math., Volume 48 (1995) no. 3, pp. 235-276

[2] G.Q. Chen; C.D. Levermore; T.P. Liu Hyperbolic conservation laws with stiff relaxation terms and entropy, Commun. Pure Appl. Math., Volume 47 (1994) no. 6, pp. 787-830

[3] R. Natalini Convergence to equilibrium for the relaxation approximations of conservation laws, Commun. Pure Appl. Math., Volume 49 (1996) no. 8, pp. 795-823

[4] C. Mascia Twenty-eight years with “Hyperbolic conservation laws with relaxation”, Acta Math. Sci., Volume 35 (2015) no. 4, pp. 807-831 (ISSN: 0252-9602) | DOI

[5] P.J. Dellar An interpretation and derivation of the lattice Boltzmann method using Strang splitting, Comput. Math. Appl., Volume 65 (2013) no. 2, pp. 129-141

[6] D. Coulette; E. Franck; P. Helluy; M. Mehrenberger; L. Navoret Palindromic discontinuous Galerkin method for kinetic equations with stiff relaxation (arXiv preprint) | arXiv

[7] D. Coulette; E. Franck; P. Helluy; M. Mehrenberger; L. Navoret Palindromic discontinuous Galerkin method, International Conference on Finite Volumes for Complex Applications, Springer, 2017, pp. 171-178

[8] T. Jahnke; C. Lubich Error bounds for exponential operator splittings, BIT Numer. Math., Volume 40 (2000) no. 4, pp. 735-744 (ISSN: 1572-9125) | DOI

[9] S. Descombes; M. Thalhammer An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime, BIT Numer. Math., Volume 50 (2010) no. 4, pp. 729-749 (ISSN: 1572-9125) | DOI

[10] R.A. Brownlee; A.N. Gorban; J. Levesley Stability and stabilization of the lattice Boltzmann method, Phys. Rev. E, Volume 75 (2007) | DOI

[11] E. Hairer; C. Lubich; G. Wanner Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31, Springer Science & Business Media, 2006

[12] R.I. McLachlan; G.R.W. Quispel Splitting methods, Acta Numer., Volume 11 (2002), pp. 341-434

[13] F. Dubois Equivalent partial differential equations of a lattice Boltzmann scheme, Comput. Math. Appl., Volume 55 (2008) no. 7, pp. 1441-1449

[14] B. Sportisse An analysis of operator splitting techniques in the stiff case, J. Comput. Phys., Volume 161 (2000) no. 1, pp. 140-168

[15] S. Chen; G.D. Doolen Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., Volume 30 (1998) no. 1, pp. 329-364

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☆ This work has been supported by ANR EXAMAG project, ANR-15-SPPE-0002.

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