Nonlinear fourth order Taylor expansion of lattice Boltzmann schemes

Article type: Research Article

Authors: Dubois, François^{a; b; *; **}

Affiliations: [a] Laboratoire de Mathématiques d’Orsay, Faculté des Sciences d’Orsay, Université Paris-Saclay, France. E-mail: francois.dubois@universite-paris-saclay.fr | [b] Conservatoire National des Arts et Métiers, LMSSC Laboratory, Paris, France

Correspondence: [**] Corresponding author. E-mail: francois.dubois@universite-paris-saclay.fr.

Note: [*] This contribution has been first presented at the Institut Henri Poincaré (Paris, France) the 10 January 2018. Edition 11 January 2021.

Abstract: We propose a formal expansion of multiple relaxation times lattice Boltzmann schemes in terms of a single infinitesimal numerical variable. The result is a system of partial differential equations for the conserved moments of the lattice Boltzmann scheme. The expansion is presented in the nonlinear case up to fourth order accuracy. The asymptotic corrections of the nonconserved moments are developed in terms of equilibrium values and partial differentials of the conserved moments. Both expansions are coupled and conduct to explicit compact formulas. The new algebraic expressions are validated with previous results obtained with this framework. The example of isothermal D2Q9 lattice Boltzmann scheme illustrates the theoretical framework.

Keywords: Partial differential equations, asymptotic analysis

DOI: 10.3233/ASY-211692

Journal: Asymptotic Analysis, vol. 127, no. 4, pp. 297-337, 2022