Equivalent partial differential equations of a lattice Boltzmann scheme

François Dubois

doi:10.1016/j.camwa.2007.08.003

paper

Equivalent partial differential equations of a lattice Boltzmann scheme

arXiv:1806.03181 · doi:10.1016/j.camwa.2007.08.003

Abstract

We show that when we formulate the lattice Boltzmann equation with a small time step $Δ$ t and an associated space scale $Δ$ x, a Taylor expansion joined with the so-called equivalent equation methodology leads to establish macroscopic fluid equations as a formal limit. We recover the Euler equations of gas dynamics at the first order and the compressible Navier-Stokes equations at the second order. 1) Discrete geometry $∙$ We denote by d the dimension of space and by L a regular d-dimensional lattice. Such a lattice is composed by a set L 0 of nodes or vertices and a set L 1 of links or edges between two vertices. From a practical point of view, given a vertex x, there exists a set V (x) of neighbouring nodes, including the node x itself. We consider here that the lattice L is parametrized by a space step $Δ$ x > 0. For the fundamental example called D2Q9 (see e.g. Lallemand and Luo, 2000), the set V (x) is given with the help of the family of vectors (e j) 0 $\leq$ j $\leq$ J defined by J = 8, (1.1) e j = 0 0 , 1 0 , 0 1 , --1 0 , 0 --1 , 1 1 , --1 1 , --1 --1 , 1 --1 and the vicinity (1.2) V (x) = { x + $Δ$ x e j , 0 $\leq$ j $\leq$ J } .

Equivalent partial differential equations of a lattice Boltzmann scheme

Abstract

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