Traditionally, time integration methods within multiphysics simulations have been chosen to cater to the most restrictive dynamics, sometimes at a great computational cost. Multirate integrators accurately and efficiently solve systems of ordinary differential equations that exhibit different time scales using two or more time steps. In this thesis, we explore three classes of time integrators that can be classified as one-step multi-stage multirate methods for which the slow dynamics are evolved using a traditional one step scheme and the fast dynamics are solved through a sequence of modified initial value problems. Practically, the fast dynamics are subcycled using a small time step and any time integration scheme of sufficient order. The overall contributions of this thesis fall into two main categories. First, we focus on the derivation of a novel class of integrators which we call implicit-explicit multirate infinitesimal generalized-structure additive Runge-Kutta (IMEX-MRI-GARK) methods. We present third and fourth order conditions for IMEX-MRI-GARK methods, consider their stability properties, and apply our derived methods to several test problems. In the second part, we discuss the numerical implementation of recently developed multirate exponential Runge-Kutta (MERK) and multirate exponential Rosenbrock (MERB) methods and their application to various test problems. MERK and MERB methods are to date some of the highest order multirate methods, with orders of convergence up to fifth and sixth order respectively. We discuss our selection of test problems for exercising these methods, present ways to experimentally determine an optimal ratio between the slow and fast time scales, and compare the performance of several multirate methods.

Degree Date

Spring 5-15-2021

Document Type


Degree Name





Daniel R. Reynolds

Subject Area

Mathematics, Applied, Mathematics, General/Other

Number of Pages




Creative Commons License

Creative Commons Attribution-Noncommercial 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License