Implicit-explicit multirate infinitesimal GARK 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 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 generalized-structure additive Runge--Kutta (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

High-order Relaxed Multirate Infinitesimal Step Methods for Multiphysics Applications

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

Adaptive Multirate Infinitesimal Time Integration

As multiphysics simulations grow in complexity and application scientists desire more accurate results, computational costs increase greatly. Time integrators typically cater to the most restrictive physical processes of a given simulation\add{,} which can be unnecessarily computationally expensive for the less restrictive physical processes. Multirate time integrators are a way to combat this increase in computational costs by efficiently solving systems of ordinary differential equations that contain physical processes which evolve at different rates by assigning different time step sizes to the different processes. Adaptivity is a technique for further increasing efficiency in time integration by automatically growing and shrinking the time step size to be as large as possible to achieve a solution accurate to a prescribed tolerance value. Adaptivity requires a time step controller, an algorithm by which the time step size is changed between steps, and benefits from an integrator with an embedding, an efficient way of estimating the error arising from each step of the integrator. In this thesis, we develop these required aspects for multirate infinitesimal time integrators, a subclass of multirate time integrators which allow for great flexibility in the treatment of the processes that evolve at the fastest rates. First, we derive the first adaptivity controllers designed specifically for multirate infinitesimal methods, and we discuss aspects of their computational implementation. Then, we derive a new class of efficient, flexible multirate infinitesimal time integrators which we name implicit-explicit multirate infinitesimal stage-restart (IMEX-MRI-SR) methods. We derive conditions guaranteeing up to fourth-order accuracy of IMEX-MRI-SR methods, explore their stability properties, provide example methods of orders two through four, and discuss their performance. Finally, we derive new instances of the class of implicit-explicit multirate infinitesimal generalized-structure additive Runge-Kutta methods, developed by Chinomona and Reynolds (2022), with embeddings and explore their stability properties and performance