Multirate Linearly-Implicit GARK Schemes

TL;DR

This paper introduces a comprehensive framework for linearly implicit multirate time integration methods, enabling efficient and stable solutions for systems with components evolving at different rates, including new flexible schemes and analysis of coupling strategies.

Contribution

It develops the MR-GARK-ROS/ROW framework for linearly implicit multirate methods, including order conditions, coupling strategies, and infinitesimal step methods, unifying and extending existing approaches.

Findings

Framework includes existing Rosenbrock methods as special cases.

Analyzes coupling strategies for efficiency and stability.

Constructs multirate infinitesimal step methods with high flexibility.

Abstract

Many complex applications require the solution of initial-value problems where some components change fast, while others vary slowly. Multirate schemes apply different step sizes to resolve different components of the system, according to their dynamics, in order to achieve increased computational efficiency. The stiff components of the system, fast or slow, are best discretized with implicit base methods in order to ensure numerical stability. To this end, linearly implicit methods are particularly attractive as they solve only linear systems of equations at each step. This paper develops the Multirate GARK-ROS/ROW (MR-GARK-ROS/ROW) framework for linearly-implicit multirate time integration. The order conditions theory considers both exact and approximative Jacobians. The effectiveness of implicit multirate methods depends on the coupling between the slow and fast computations; an array of efficient coupling strategies and the resulting numerical schemes are analyzed. Multirate infinitesimal step linearly-implicit methods, that allow arbitrarily small micro-steps and offer extreme computational flexibility, are constructed. The new unifying framework includes existing multirate Rosenbrock(-W) methods as particular cases, and opens the possibility to develop new classes of highly effective linearly implicit multirate integrators.