Implicit-explicit multirate infinitesimal methods

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 flexibility at the slow time scale, 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 solver. We name the proposed class as implicit-explicit multirate infinitesimal (IMEX-MRI) methods. In addition to defining these methods, we prove that they may be viewed as specific instances of generalized-structure additive Runge--Kutta (GARK) methods, and derive a set of order conditions on the IMEX-MRI coefficients to guarantee both third and fourth order accuracy for the overall multirate method. Additionally, we provide three specific IMEX-MRI methods, two of order three and one of order four. We conclude with numerical simulations demonstrating their predicted convergence rates on two multirate test problems, and compare their efficiency against legacy IMEX multirate methods.