Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves

TL;DR

This paper introduces a dual entropy approach for lattice Boltzmann schemes, providing a new mathematical framework that enhances stability and applicability for nonlinear wave simulations, especially in acoustics.

Contribution

It develops a dual entropy method for lattice Boltzmann schemes, establishing a new stability framework and demonstrating its effectiveness for nonlinear waves and acoustics.

Findings

Derives three equilibrium distributions for D1Q3 schemes

Achieves stable numerical simulations of nonlinear shocks and rarefactions

Extends the approach to nonlinear acoustics but not to shallow water equations

Abstract

We follow the mathematical framework proposed by Bouchut and present in this contribution a dual entropy approach for determining equilibrium states of a lattice Boltzmann scheme. This method is expressed in terms of the dual of the mathematical entropy relative to the underlying conservation law. It appears as a good mathematical framework for establishing a "H-theorem" for the system of equations with discrete velocities. The dual entropy approach is used with D1Q3 lattice Boltzmann schemes for the Burgers equation. It conducts to the explicitation of three different equilibrium distributions of particles and induces naturally a nonlinear stability condition. Satisfactory numerical results for strong nonlinear shocks and rarefactions are presented. We prove also that the dual entropy approach can be applied with a D1Q3 lattice Boltzmann scheme for systems of linear and nonlinear acoustics and we present a numerical result with strong nonlinear waves for nonlinear acoustics. We establish also a negative result: with the present framework, the dual entropy approach cannot be used for the shallow water equations.