Nonlinear fourth order Taylor expansion of lattice Boltzmann schemes

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.

Finite Difference formulation of any lattice Boltzmann scheme

“…The proof is the same than that of Proposition 4 by taking advantage of Corollary 5. We show in another contribution [1] that the result of Proposition 6 is the right one to bridge between the consistency analysis of Finite Difference schemes and the Taylor expansions on the lattice Boltzmann schemes for N ≥ 1 proposed by [17].…”

“…where for the first time, the matrices have entries in a commutative ring, see [19] and [6], instead than in the field R. The set M q (D d ∆x ) of square matrices of size q with entries belonging to D d ∆x forms a ring under the usual operations between matrices. Even if D d ∆x is commutative from Proposition 2, M q (D d ∆x ) is not commutative for q ≥ 2, as for real matrices and matrices of first-order differential operators [17].…”

“…The stream phase Equation ( 5) can be rewritten in a non-diagonal form in the space of moments as done by [17,20] by introducing the matrix T := M diag(T c1 ∆x , . .…”

“…Although this way of proceeding is highly beneficial from a computational perspective, it yields a deficient structure to construct a clear and rigorous notion of consistency with respect to the N target equations, as well as a rigorous theory of stability. Indeed, only formal procedures, either based on the Chapman-Enskog expansion [8] or on the equivalent equations by Dubois [15,17] are currently available to study the consistency of lattice Boltzmann schemes. As far as stability is concerned, most of the studies rely on the linear stability analysis of the eigenvalues of the system, see [4,37].…”

“…Their focus is different than ours since they adopt a purely algorithmic approach rather than a precise algebraic characterization of lattice Boltzmann schemes. We actually provide more insight into the bound on the number of time steps of the corresponding Finite Difference scheme and our formalism, based on polynomials, aims at providing a direct link with the classical tools for the stability analysis and allows to establish a link with the Taylor expansions from [17], as introduced in [1]. In [21], the authors rely on a decomposition of the scheme using an hollow matrix 2 yielding an equivalent form of the scheme with the diagonal non-equilibrium part, after a finite number of steps of their algorithm.…”