Implicit multirate GARK methods

TL;DR

This paper introduces implicit multirate GARK methods for efficiently solving stiff ODE systems with multiple time scales, balancing stability and computational cost through novel coupling strategies and stability analysis.

Contribution

It develops new implicit multirate GARK methods up to fourth order, analyzes stability limitations, and demonstrates their effectiveness with numerical tests.

Findings

New implicit multirate methods up to fourth order

General stability analysis and fundamental limitations identified

Numerical tests confirm accuracy and efficiency improvements

Abstract

This work considers multirate generalized-structure additively partitioned Runge-Kutta (MrGARK) methods for solving stiff systems of ordinary differential equations (ODEs) with multiple time scales. These methods treat different partitions of the system with different timesteps for a more targeted and efficient solution compared to monolithic single rate approaches. With implicit methods used across all partitions, methods must find a balance between stability and the cost of solving nonlinear equations for the stages. In order to characterize this important trade-off, we explore multirate coupling strategies, problems for assessing linear stability, and techniques to efficiently implement Newton iterations for stage equations. Unlike much of the existing multirate stability analysis which is limited in scope to particular methods, we present general statements on stability and describe fundamental limitations for certain types of multirate schemes. New implicit multirate methods up to fourth order are derived, and their accuracy and efficiency properties are verified with numerical tests.