Fast parallel solution of fully implicit Runge-Kutta and discontinuous Galerkin in time for numerical PDEs, Part I: the linear setting

Fully implicit Runge-Kutta (IRK) methods have many desirable properties as time integration schemes in terms of accuracy and stability, but are rarely used in practice with numerical PDEs due to the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic framework for the fast, parallel solution of the systems of equations that arise from IRK methods applied to linear numerical PDEs (without algebraic constraints). This framework also naturally applies to discontinuous Galerkin discretizations in time. The new method can be used with arbitrary existing preconditioners for backward Euler-type time stepping schemes, and is amenable to the use of three-term recursion Krylov methods when the underlying spatial discretization is symmetric. Under quite general assumptions on the spatial discretization that yield stable time integration, the preconditioned operator is proven to have conditioning ∼ O(1), with only weak dependence on number of stages/polynomial order; for example, the preconditioned operator for 10th-order Gauss integration has condition number less than two. The new method is demonstrated to be effective on various high-order finite-difference and finite-element discretizations of linear parabolic and hyperbolic problems, demonstrating fast, scalable solution of up to 10th order accuracy. In several cases, the new method can achieve 4th-order accuracy using Gauss integration with roughly half the number of preconditioner applications as required using standard SDIRK techniques. ∗BSS was supported by Lawrence Livermore National Laboratory under contract B639443, and as a Nicholas C. Metropolis Fellow under the Laboratory Directed Research and Development program of Los Alamos National Laboratory. OAK acknowledges the support of an Australian Government Research Training Program (RTP) Scholarship. †Theoretical Division, Los Alamos National Laboratory, U.S.A. (southworth@lanl.gov), http:// orcid.org/0000-0002-0283-4928 ‡School of Mathematics, Monash University, Australia (oliver.krzysik@monash.edu), https:// orcid.org/0000-0001-7880-6512 §Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, U.S.A. (pazner1@llnl.gov) ¶Department of Applied Mathematics, University of Waterloo, Waterloo, Canada (hdesterck@ uwaterloo.ca) 1 ar X iv :2 10 1. 00 51 2v 2 [ m at h. N A ] 7 J an 2 02 1

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