Residual equilibrium schemes for time dependent partial differential equations

TL;DR

This paper introduces residual equilibrium schemes that accurately preserve steady states and achieve high order accuracy for time-dependent PDEs, including nonlinear diffusion and kinetic equations, inspired by micro-macro decomposition methods.

Contribution

The paper proposes a novel class of residual equilibrium schemes that maintain steady states and attain high order accuracy for various PDEs, extending to systems with source terms.

Findings

Schemes successfully preserve steady states in tested PDEs.

High order accuracy achieved in nonlinear diffusion and kinetic equations.

Extension to conservation law systems demonstrated.

Abstract

Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed.