# A new class of high-order methods for multirate differential equations

## TL;DR

This paper introduces a new class of high-order multirate exponential Runge--Kutta methods (MERK) for efficiently solving multirate differential equations, with rigorous convergence analysis and numerical validation.

## Contribution

It develops MERK methods based on explicit exponential Runge--Kutta schemes, providing a novel multirate integration approach with proven convergence properties.

## Key findings

- MERK methods achieve high-order accuracy (orders 3-5).
- Numerical tests confirm the efficiency and convergence of MERK methods.
- Fast problems can be solved with order p-1 methods, final with order p.

## Abstract

This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. The proposed methods are based on a specific subset of explicit one-step exponential integrators. More precisely, starting from an explicit exponential Runge--Kutta method of the appropriate form, we derive a multirate algorithm to approximate the action of the matrix exponential through the definition of modified "fast" initial-value problems. These fast problems may be solved using any viable solver, enabling multirate simulations through use of a subcycled method. Due to this structure, we name these Multirate Exponential Runge--Kutta (MERK) methods. In addition to showing how MERK methods may be derived, we provide rigorous convergence analysis, showing that for an overall method of order $p$, the fast problems corresponding to internal stages may be solved using a method of order $p-1$, while the final fast problem corresponding to the time-evolved solution must use a method of order $p$. Numerical simulations are then provided to demonstrate the convergence and efficiency of MERK methods with orders three through five on a series of multirate test problems.