SAV-based entropy-dissipative schemes for a class of kinetic equations

TL;DR

This paper develops entropy-dissipative numerical schemes for kinetic equations using the scalar auxiliary variable approach, ensuring positivity and robustness, validated through extensive numerical tests on Boltzmann and Landau equations.

Contribution

It introduces novel SAV-based schemes that are entropy-dissipative and positivity-preserving for kinetic equations, including specific schemes for Boltzmann and Landau equations.

Findings

Schemes are proven to be entropy-dissipative and accurate.

Positivity-preserving schemes enhance robustness.

Numerical examples confirm theoretical properties.

Abstract

We introduce novel entropy-dissipative numerical schemes for a class of kinetic equations, leveraging the recently introduced scalar auxiliary variable (SAV) approach. Both first and second order schemes are constructed. Since the positivity of the solution is closely related to entropy, we also propose positivity-preserving versions of these schemes to ensure robustness, which include a scheme specially designed for the Boltzmann equation and a more general scheme using Lagrange multipliers. The accuracy and provable entropy-dissipation properties of the proposed schemes are validated for both the Boltzmann equation and the Landau equation through extensive numerical examples.