Stability of a bidimensional relative velocity lattice Boltzmann scheme

TL;DR

This paper analyzes the stability of a bidimensional relative velocity lattice Boltzmann scheme, comparing different velocity choices and their effects on L1 and L2 stability, with theoretical and numerical insights.

Contribution

It provides a detailed stability analysis of a four-velocity scheme with respect to the relative velocity parameter, including new stability conditions and comparisons.

Findings

Choosing relative velocity equal to advection velocity improves L2 stability.

L1 stability regions are fully characterized by relaxation parameters and advection velocity.

No hierarchy exists between the two relative velocity choices for L1 stability.

Abstract

In this contribution, we study the theoretical and numerical stability of a bidimensional relative velocity lattice Boltzmann scheme. These relative velocity schemes introduce a velocity field parameter called "relative velocity" function of space and time. They generalize the d'Humi\`eres multiple relaxation times scheme and the cascaded automaton. This contribution studies the stability of a four velocities scheme applied to a single linear advection equation according to the value of this relative velocity. We especially compare when it is equal to 0 (multiple relaxation times scheme) or to the advection velocity ("cascaded like" scheme). The comparison is made in terms of L1 and L2 stability. The L1 stability area is fully described in terms of relaxation parameters and advection velocity for the two choices of relative velocity. These results establish that no hierarchy of these two choices exists for the L1 notion. Instead, choosing the parameter equal to the advection velocity improves the numerical L2 stability of the scheme. This choice cancels some dispersive terms and improve the numerical stability on a representative test case. We theoretically strengthen these results with a weighted L2 notion of stability.