Implicit Extensions of an Explicit Multirate Runge-Kutta Scheme

TL;DR

This paper introduces an implicit extension to explicit multirate Runge-Kutta schemes, enhancing their stability and applicability for adaptive mesh refinement PDEs with varying stiffness.

Contribution

It develops new implicit-explicit multirate methods of order one and two, tailored for stiff PDE discretizations with adaptive meshes.

Findings

The new methods demonstrate improved stability properties.

Numerical experiments confirm effectiveness on advection-diffusion problems.

Extensions are suitable for adaptive mesh refinement applications.

Abstract

We propose a new method that extends conservative explicit multirate methods to implicit explicit-multirate methods. We develop extensions of order one and two with different stability properties on the implicit side. The method is suitable for time-stepping adaptive mesh refinement PDE discretizations with different degrees of stiffness. A numerical example with an advection-diffusion problem illustrates the new method's properties.