Implicit-explicit multirate infinitesimal GARK methods

TL;DR

This paper introduces a new class of high-order multirate time integration methods for ODEs that support mixed implicit-explicit treatment and utilize an infinitesimal fast scale formulation, enhancing flexibility and accuracy.

Contribution

The paper develops IMEX-MRI-GARK methods that combine implicit-explicit multirate integration with an infinitesimal fast scale approach, providing new order conditions and specific methods up to fourth order.

Findings

Methods achieve predicted convergence rates

Demonstrated improved efficiency over legacy schemes

Validated accuracy with numerical simulations

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. Unlike other recent work in this area, the proposed methods support mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to allowing this slow time scale flexibility, the proposed methods utilize a so-called `infinitesimal' formulation for the fast time scale through definition of a sequence of modified `fast' initial-value problems, that may be solved using any viable algorithm. We name the proposed class as implicit-explicit multirate infinitesimal generalized-structure additive Runge--Kutta (IMEX-MRI-GARK) methods. In addition to defining these methods, we prove that they may be viewed as specific instances of GARK methods, and derive a set of order conditions on the IMEX-MRI-GARK coefficients to guarantee both third and fourth order accuracy for the overall multirate method. Additionally, we provide three specific IMEX-MRI-GARK methods, two of order three and one of order four. We conclude with numerical simulations on two multirate test problems, demonstrating the methods' predicted convergence rates and comparing their efficiency against both legacy IMEX multirate schemes and recent third and fourth order implicit MRI-GARK methods.