Fully Implicit Time Integration: Fast Iterative Solvers and Implicit-Explicit Schemes

Staff - Faculty of Informatics

Date: 14 September 2021 / 14:30 - 15:30

USI Campus EST, room D1.14, Sector D

Ben S. Southworth,  Nicholas Metropolis Fellow, Los Alamos National Laboratory

Oliver Krzysik, Will Pazner, Hans De Sterck

Fully implicit Runge Kutta (IRK) methods (Gauss, Radau, etc.) offer a number of desirable properties over their more-commonly used diagonally implicit (DIRK) counterparts. However, applying IRK methods in the context of numerical PDEs requires the solution of large block systems of equations, which has significantly limited their use in practice. In addition, additive implicit-explicit (IMEX) integration schemes are often used in practice to integrate complicated PDEs, as they simplify the implicit solve. Unfortunately, a traditional IRK formulation does not permit a natural implicit-explicit extension, as the explicit stages would be nonlinearly coupled to the implicit stages. In this talk we will address each of these challenges.
First, we introduce a theoretical and algorithmic framework for the fast solution of fully implicit systems, which also naturally applies to discontinuous Galerkin (DG) discretizations in time. The new method can be used with arbitrary existing preconditioners for backward Euler-type time stepping schemes, and is amenable to more robust nonlinear iterations than the simplified Newton approach typically considered with IRK methods. Under quite general assumptions on the spatial discretization that yield stable time integration, the preconditioned operator is proven to have O(1) conditioning, with only weak dependence on the number of stages/polynomial order. The method is demonstrated to be effective on various discretizations of parabolic and hyperbolic problems, often outperforming classical DIRK methods in terms of wallclock time.
Second, we introduce the framework of (additive) polynomial time integrators, which construct time-integration schemes based on continuous polynomials in time. We then demonstrate how this framework can be used to easily construct fully-implicit-explicit integration schemes, and focus on an IMEX-RadauIIA class of methods. Coupling with the solvers from part I, the new methods are applied to DG advection-diffusion discretizations and shown to have superior accuracy and stability compared with existing IMEX-DIRK methods.

Dr. Southworth obtained his Ph.D. in applied mathematics from the University of Colorado at Boulder in 2017. After his Ph.D., he worked as a postdoctoral Research Associate also at the University of Colorado. Since 2020 he is appointed at Los Alamos National Lab as Nicholas C. Metropolis Postdoc Fellow. His research interests include algebraic multigrid, block preconditioners, space-time discretizations, parallel algorithms and high-order meshes for radiative transport, parallel-in-time methods, and more recently machine learning approaches.

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