Multirate Exponential Rosenbrock Methods

TL;DR

This paper introduces Multirate Exponential Rosenbrock (MERB) methods for high-order accurate integration of multirate ODE systems, combining exponential Rosenbrock schemes with multirate techniques.

Contribution

It develops a new class of multirate methods based on exponential Rosenbrock schemes, achieving the highest order (up to six) for infinitesimal multirate methods and providing rigorous convergence analysis.

Findings

MERB methods achieve high-order accuracy from two to six.

Numerical simulations confirm theoretical convergence rates.

MERB methods outperform some existing high-order multirate methods.

Abstract

In this paper we propose a novel class of methods for high order accurate integration of multirate systems of ordinary differential equation initial-value problems. The proposed methods construct multirate schemes by approximating the action of matrix $\varphi$-functions within explicit exponential Rosenbrock (ExpRB) methods, thereby called Multirate Exponential Rosenbrock (MERB) methods. They consist of the solution to a sequence of modified "fast" initial-value problems, that may themselves be approximated through subcycling any desired IVP solver. In addition to proving how to construct MERB methods from certain classes of ExpRB methods, we provide rigorous convergence analysis of these methods and derive efficient MERB schemes of orders two through six (the highest order ever constructed infinitesimal multirate methods). We then present numerical simulations to confirm these theoretical convergence rates, and to compare the efficiency of MERB methods against other recently-introduced high order multirate methods.