Efficient Multigrid Reduction-in-Time for Method-of-Lines Discretizations of Linear Advection

• Published in: Journal of scientific computing Vol. 96; no. 1; p. 1
• Main Authors: De Sterck, H., Falgout, R. D., Krzysik, O. A., Schroder, J. B.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2023; Springer Nature B.V; Springer
• Subjects: Advection; advection equation; Algorithms; Approximation; Coarsening; Computational Mathematics and Numerical Analysis; Convergence; Error correction; hyperbolic PDE; Iterative methods; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; MATHEMATICS AND COMPUTING; Mathematics and Statistics; Method of lines; Methods; MGRIT; multigrid; Multigrid methods; Optimization; Parallel-in-time; parareal; Partial differential equations; Reduction; Robustness (mathematics); Runge-Kutta method; Spacetime; Theoretical; Truncation errors; Parareal; Advection equation; 65F10; Hyperbolic PDE; Multigrid; 35L03; 65M22; 65M55; MGRIT; Parallel-in-time
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-023-02223-4

Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. Here we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Runge-Kutta methods in time. Using a convergence framework we developed in previous work, we prove for a subclass of these discretizations that, if using the standard approach of rediscretizing the fine-grid problem on the coarse grid, robust MGRIT convergence with respect to CFL number and coarsening factor is not possible. This poor convergence and non-robustness is caused, at least in part, by an inadequate coarse-grid correction for smooth Fourier modes in space-time known as characteristic components. We propose an alternative coarse-grid operator that provides a better correction of these modes. This coarse-grid operator is related to previous work and uses a semi-Lagrangian discretization combined with an implicitly treated truncation error correction. Theory and numerical experiments show the proposed coarse-grid operator yields fast MGRIT convergence for many of the method-of-lines discretizations considered, including for both implicit and explicit discretizations of high order. Parallel results demonstrate speed-up over sequential time-stepping.