Entropy-Preserving and Entropy-Stable Relaxation IMEX and Multirate Time-Stepping Methods

TL;DR

This paper introduces entropy-preserving and entropy-stable IMEX and multirate Runge--Kutta methods that enhance stability and entropy properties for stiff problems, with theoretical derivations and numerical validations on Burgers' equation.

Contribution

It extends relaxation Runge--Kutta methods to IMEX and multirate schemes, enabling entropy preservation and stability in stiff systems with explicit relaxation parameters.

Findings

Methods successfully preserve or stabilize entropy in numerical simulations.

Explicit relaxation parameters derived for quadratic entropy functions.

Numerical experiments confirm entropy behavior and compare approaches with shocks.

Abstract

We propose entropy-preserving and entropy-stable partitioned Runge--Kutta (RK) methods. In particular, we extend the explicit relaxation Runge--Kutta methods to IMEX--RK methods and a class of explicit second-order multirate methods for stiff problems arising from scale-separable or grid-induced stiffness in a system. The proposed approaches not only mitigate system stiffness but also fully support entropy-preserving and entropy-stability properties at a discrete level. The key idea of the relaxation approach is to adjust the step completion with a relaxation parameter so that the time-adjusted solution satisfies the entropy condition at a discrete level. The relaxation parameter is computed by solving a scalar nonlinear equation at each timestep in general; however, as for a quadratic entropy function, we theoretically derive the explicit form of the relaxation parameter and numerically confirm that the relaxation parameter works the Burgers equation. Several numerical results for ordinary differential equations and the Burgers equation are presented to demonstrate the entropy-conserving/stable behavior of these methods. We also compare the relaxation approach and the incremental direction technique for the Burgers equation with and without a limiter in the presence of shocks.