Multirate GARK Schemes for Multiphysics Problems

Abstract

Multirate GARK schemes define a multirate extension of GARK schemes, generalized additive Runge-Kutta schemes. In contrast to additive schemes, GARK schemes allow for different stage values as arguments of different components of the right hand side. They introduce additional flexibility when compared to traditional partitioned Runge-Kutta methods, and therefore offer additional opportunities for the development of flexible solvers for systems with multiple scales, or driven by multiple physical processes.

Consequently, multirate GARK schemes allow for exploiting multirate behaviour in both the right-hand sides and in the components in a rather general setting, and are thus especially useful for coupled problems in a multiphysics setting. We apply MGARK schemes to a benchmark example from thermal-electrical coupling, characterized by a slow and fast part with a stiff and non-stiff characteristic, resp. We test two MGARK schemes: (a) an IMEX method, which completely utilizes the dynamics and differing stability properties of the coupled subsystem; and (b) a fully implicit schemes, which inherits the stability properties from both underlying schemes without any coupling constraint.