In this work, we consider numerical methods for integrating multirate ordinary differential equations. We are interested in the development of new multirate methods with good stability properties and improved efficiency over existing methods. We discuss the development of multirate methods, particularly focusing on those that are based on Runge-Kutta theory. We introduce the theory of Generalized Additive Runge-Kutta methods proposed by Sandu and Günther. We also introduce the theory of Recursive Flux Splitting Multirate Methods with Sub-cycling described by Schlegel, as well as the Multirate Infinitesimal Step methods this work is based on. We propose a generic structure called Flexible Multirate Generalized-Structure Additively-Partitioned Runge-Kutta methods which allows for optimization and more rigorous analysis. We also propose a specific class of higher-order methods, called Relaxed Multirate Infinitesimal Step Methods. We will leverage GARK theories to develop new theory about the stability and accuracy of these new methods.

Degree Date

Fall 2017

Document Type





Daniel R. Reynolds



Creative Commons License

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