Time Step Expansions and the Invariant Manifold Approach to Lattice Boltzmann Models

TL;DR

This paper introduces a novel approach to derive macroscopic dynamics of lattice Boltzmann models by focusing on invariant manifolds and time step expansions, avoiding traditional approximations.

Contribution

It presents a new method starting from fully discrete schemes to directly obtain macroscopic equations, including second-order effects, and analyzes stability conditions.

Findings

Derived macroscopic dynamics up to second order in time step.

Identified the deviation from Navier-Stokes equations due to velocity discretization.

Established stability conditions through short wave perturbation analysis.

Abstract

The classical method for deriving the macroscopic dynamics of a lattice Boltzmann system is to use a combination of different approximations and expansions. Usually a Chapman-Enskog analysis is performed, either on the continuous Boltzmann system, or its discrete velocity counterpart. Separately a discrete time approximation is introduced to the discrete velocity Boltzmann system, to achieve a practically useful approximation to the continuous system, for use in computation. Thereafter, with some additional arguments, the dynamics of the Chapman-Enskog expansion are linked to the discrete time system to produce the dynamics of the completely discrete scheme. In this paper we put forward a different route to the macroscopic dynamics. We begin with the system discrete in both velocity space and time. We hypothesize that the alternating steps of advection and relaxation, common to all lattice Boltzmann schemes, give rise to a slow invariant manifold. We perform a time step expansion of the discrete time dynamics using the invariance of the manifold. Finally we calculate the dynamics arising from this system. By choosing the fully discrete scheme as a starting point we avoid mixing approximations and arrive at a general form of the microscopic dynamics up to the second order in the time step. We calculate the macroscopic dynamics of two commonly used lattice schemes up to the first order, and hence find the precise form of the deviation from the Navier-Stokes equations in the dissipative term, arising from the discretization of velocity space. Finally we perform a short wave perturbation on the dynamics of these example systems, to find the necessary conditions for their stability.