Equivalent partial differential equations of a lattice Boltzmann scheme

Equivalent partial differential equations of a lattice Boltzmann scheme

François Dubois (1, 2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 LMSSC - Laboratoire de Mécanique des Structures et des Systèmes Couplés

We show that when we formulate the lattice Boltzmann equation with a small time step ∆t and an associated space scale ∆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 ∆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≤j≤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 + ∆x e j , 0 ≤ j ≤ J } .

Domaines: Analyse numérique [math.NA]; Mécanique des fluides [physics.class-ph]

François Dubois. Equivalent partial differential equations of a lattice Boltzmann scheme. Computers & Mathematics with Applications, 2008, 55 (7), pp.1441 - 1449. ⟨10.1016/j.camwa.2007.08.003⟩. ⟨hal-01809947⟩