Equivalent partial differential equations of a lattice Boltzmann scheme
For researchers in lattice Boltzmann methods, this provides a rigorous theoretical justification for the connection between discrete lattice Boltzmann schemes and continuum fluid dynamics.
The paper derives macroscopic fluid equations (Euler and compressible Navier-Stokes) from the lattice Boltzmann equation using Taylor expansion and equivalent equation methodology, showing that the lattice Boltzmann scheme recovers these equations as a formal limit at first and second order in the time step.
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 $\bullet$ 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$\le$j$\le$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 $\le$ j $\le$ J } .