Efficient implementation of partitioned stiff exponential Runge-Kutta methods

TL;DR

This paper develops efficient partitioned exponential Runge-Kutta methods tailored for multiphysics systems, enabling accurate simulations by computing matrix functions of individual components' Jacobians rather than the full system.

Contribution

It introduces structured implementations of exponential Runge-Kutta methods that preserve order and efficiency for partitioned multiphysics differential equations.

Findings

Methods retain full order for various partitionings.

Application to reaction-diffusion problems demonstrates effectiveness.

Computational savings by focusing on component Jacobians.

Abstract

Multiphysics systems are driven by multiple processes acting simultaneously, and their simulation leads to partitioned systems of differential equations. This paper studies the solution of partitioned systems of differential equations using exponential Runge-Kutta methods. We propose specific multiphysics implementations of exponential Runge-Kutta methods satisfying stiff order conditions that were developed in [Hochbruck et al., SISC, 1998] and [Luan and Osterman, JCAM, 2014]. We reformulate stiffly--accurate exponential Runge--Kutta methods in a way that naturally allows of the structure of multiphysics systems, and discuss their application to both component and additively partitioned systems. The resulting partitioned exponential methods only compute matrix functions of the Jacobians of individual components, rather than the Jacobian of the full, coupled system. We derive modified formulations of particular methods of order two, three and four, and apply them to solve a partitioned reaction-diffusion problem. The proposed methods retain full order for several partitionings of the discretized problem, including by components and by physical processes.