Flexible Time Integration Methods for Multiphysics Partial Differential Equation Systems

Abstract: In recent years, computational simulations have rapidly evolved in complexity (high order discretizations, spatial adaptivity, additional physical processes), placing ever larger strains on the time integration methods on which they rely. High spatial order necessitates comparably high order time integration. Spatial adaptivity and multiphysics processes give rise to subsets of the solution that evolve at differing time scales, or to simulations that combine nonstiff but highly nonlinear processes with others that may be highly stiff but that are frequently linear. In this talk, I will discuss recent work on advanced time integration methods that allow extreme flexibility in the use of different techniques to distinct physical processes, while still allowing high orders of accuracy. I will primarily focus on the newly-developed IMEX-MRI-GARK methods (Chinomona and R., 2021), and their implementation in the ARKODE library within SUNDIALS, but I will additionally point out other recent related work.

Speaker’s Bio: Daniel Reynolds is Professor and Chair of the Department of Mathematics at Southern Methodist University (SMU). His research focuses on the development and application of robust time integrators and iterative nonlinear and linear solvers for large-scale multiphysics systems -- simulations comprised of multiple interacting physical processes. Reynolds is the lead developer of the ARKODE library, a collection of solvers for adaptive implicit-explicit and multirate time integration methods for large-scale systems of differential equations, and is a senior developer for the larger SUNDIALS suite of time integration and nonlinear solver libraries (which includes ARKODE). He has been an active member of the Department of Energy Scientific Discovery through Advanced Computing program since 2003, where he has focused on nonlinear solvers and time integration methods as part of the TOPS project and FASTMath institute.

Prior to joining SMU in 2008, Reynolds held postdoctoral research positions in the Department of Mathematics and the Center for Astrophysics and Space Sciences at the University of California, San Diego (2005-2008), and in the Center for Applied Scientific Computing at Lawrence Livermore National Laboratory (2003-2005).  He received a PhD in Computational and Applied Mathematics from Rice University in 2003, and a BA in Mathematics from Southwestern University in 1998.