Convergence analysis for parallel‐in‐time solution of hyperbolic systems

Summary Parallel‐in‐time algorithms have been successfully employed for reducing time‐to‐solution of a variety of partial differential equations, especially for diffusive (parabolic‐type) equations. A major failing of parallel‐in‐time approaches to date, however, is that most methods show instabilit...

Full description

Saved in:
Bibliographic Details
Published inNumerical linear algebra with applications Vol. 27; no. 1
Main Authors De Sterck, Hans, Friedhoff, Stephanie, Howse, Alexander J. M., MacLachlan, Scott P.
Format Journal Article
LanguageEnglish
Published Oxford Wiley Subscription Services, Inc 01.01.2020
Subjects
Algebra
Algorithms
Coarsening
Convergence
Elasticity
Error analysis
Fourier analysis
hyperbolic PDEs
Hyperbolic systems
mode analysis
Multigrid methods
multigrid reduction in time
parallel‐in‐time
parareal
Partial differential equations
Reduction
Online AccessGet full text

Cover

More Information
Summary:Summary Parallel‐in‐time algorithms have been successfully employed for reducing time‐to‐solution of a variety of partial differential equations, especially for diffusive (parabolic‐type) equations. A major failing of parallel‐in‐time approaches to date, however, is that most methods show instabilities or poor convergence for hyperbolic problems. This paper focuses on the analysis of the convergence behavior of multigrid methods for the parallel‐in‐time solution of hyperbolic problems. Three analysis tools are considered that differ, in particular, in the treatment of the time dimension: (a) space–time local Fourier analysis, using a Fourier ansatz in space and time; (b) semi‐algebraic mode analysis, coupling standard local Fourier analysis approaches in space with algebraic computation in time; and (c) a two‐level reduction analysis, considering error propagation only on the coarse time grid. In this paper, we show how insights from reduction analysis can be used to improve feasibility of the semi‐algebraic mode analysis, resulting in a tool that offers the best features of both analysis techniques. Following validating numerical results, we investigate what insights the combined analysis framework can offer for two model hyperbolic problems, the linear advection equation in one space dimension and linear elasticity in two space dimensions.
ISSN:1070-5325
1099-1506
DOI:10.1002/nla.2271