Joint Applied Mathematics and Statistics Colloquium: Daniel Reynolds (SMU)

Abstract: In recent years computational simulations have rapidly evolved in complexity (high order discretizations, spatial adaptivity, wide arrays of 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/nonlinear processes with others that may be highly stiff but that are frequently linear.  In this talk, I will discuss recent work on time integration methods that allow the flexibility to apply different techniques to distinct physical processes.  While techniques for flexible time integration have existed for some time, including additive Runge--Kutta ImEx, multirate (a.k.a. multiple time stepping), and operator-splitting methods, there have been comparably few that combine these types of flexibility into a single family, while also supporting high orders of accuracy and temporal adaptivity.  In this talk, I focus on the newly developed IMEX-MRI-GARK (Chinomona and R., 2022) and IMEX-MRI-SR (Fish, R. and Roberts, 2023) families of methods, along with novel techniques for time adaptivity in multirate infinitesimal time integration methods (Fish and R., 2023).  In addition to discussing these methods themselves, I'll also present our work implementing and releasing these as part of the open-source SUNDIALS library.