A Class of Multirate Infinitesimal GARK Methods

TL;DR

This paper introduces a new family of multirate infinitesimal GARK methods designed to efficiently solve differential equations with multiple time scales, extending previous hybrid approaches with theoretical analysis and practical schemes up to order four.

Contribution

The paper develops a comprehensive family of MRI-GARK schemes, including order conditions, stability analysis, and explicit/implicit methods, advancing multirate integration techniques.

Findings

Theoretical order conditions and stability analyses are established.

Explicit and implicit methods of up to order four are constructed.

Numerical experiments validate the theoretical properties.

Abstract

Differential equations arising in many practical applications are characterized by multiple time scales. Multirate time integration seeks to solve them efficiently by discretizing each scale with a different, appropriate time step, while ensuring the overall accuracy and stability of the numerical solution. In a seminal paper Knoth and Wolke (APNUM, 1998) proposed a hybrid solution approach: discretize the slow component with an explicit Runge-Kutta method, and advance the fast component via a modified fast differential equation. The idea led to the development of multirate infinitesimal step (MIS) methods by Wensch et al. (BIT, 2009.)G\"{u}nther and Sandu (BIT, 2016) explained MIS schemes as a particular case of multirate General-structure Additive Runge-Kutta (MR-GARK) methods. The hybrid approach offers extreme flexibility in the choice of the numerical solution process for the fast component. This work constructs a family of multirate infinitesimal GARK schemes (MRI-GARK) that extends the hybrid dynamics approachin multiple ways. Order conditions theory and stability analyses are developed, and practical explicit and implicit methods of up to order four are constructed. Numerical results confirm the theoretical findings. We expect the new MRI-GARK family to be most useful for systems of equations with widely disparate time scales, where the fast process is dispersive, and where the influence of the fast component on the slow dynamics is weak.