Step size control for explicit relaxation Runge-Kutta methods preserving invariants

TL;DR

This paper explores how to integrate relaxation techniques with step size control in explicit Runge-Kutta methods to preserve invariants in differential equations, enhancing solution quality in numerical simulations.

Contribution

It introduces a novel approach combining relaxation methods with local error-based step size control for invariant-preserving explicit Runge-Kutta schemes.

Findings

Successful preservation of invariants in numerical experiments

Improved accuracy and stability in solving ODEs and PDEs

Effective step size adaptation maintaining invariant properties

Abstract

Many time-dependent differential equations are equipped with invariants. Preserving such invariants under discretization can be important, e.g., to improve the qualitative and quantitative properties of numerical solutions. Recently, relaxation methods have been proposed as small modifications of standard time integration schemes guaranteeing the correct evolution of functionals of the solution. Here, we investigate how to combine these relaxation techniques with efficient step size control mechanisms based on local error estimates for explicit Runge-Kutta methods. We demonstrate our results in several numerical experiments including ordinary and partial differential equations.