Optimizing multigrid reduction-in-time and Parareal coarse-grid operators for linear advection

De Sterck, Hans; Falgout, Robert D.; Friedhoff, Stephanie; Krzysik, Oliver A.; MacLachlan, Scott P.

doi:10.1002/nla.2367

Title: Optimizing multigrid reduction-in-time and Parareal coarse-grid operators for linear advection

Journal Article · Tue Mar 09 00:00:00 EST 2021 · Numerical Linear Algebra with Applications

DOI:https://doi.org/10.1002/nla.2367· OSTI ID:1819027

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Univ. of Waterloo, ON (Canada)
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing (CASC)
Bergische Univ., Wuppertal (Germany)
Monash Univ., Clayton, VIC (Australia)
Univ. of Newfoundland, St. John's, NL (Canada)

Abstract Parallel‐in‐time methods, such as multigrid reduction‐in‐time (MGRIT) and Parareal, provide an attractive option for increasing concurrency when simulating time‐dependent partial differential equations (PDEs) in modern high‐performance computing environments. While these techniques have been very successful for parabolic equations, it has often been observed that their performance suffers dramatically when applied to advection‐dominated problems or purely hyperbolic PDEs using standard rediscretization approaches on coarse grids. In this paper, we apply MGRIT or Parareal to the constant‐coefficient linear advection equation, appealing to existing convergence theory to provide insight into the typically nonscalable or even divergent behavior of these solvers for this problem. To overcome these failings, we replace rediscretization on coarse grids with improved coarse‐grid operators that are computed by applying optimization techniques to approximately minimize error estimates from the convergence theory. One of our main findings is that, in order to obtain fast convergence as for parabolic problems, coarse‐grid operators should take into account the behavior of the hyperbolic problem by tracking the characteristic curves. Our approach is tested for schemes of various orders using explicit or implicit Runge–Kutta methods combined with upwind‐finite‐difference spatial discretizations. In all cases, we obtain scalable convergence in just a handful of iterations, with parallel tests also showing significant speed‐ups over sequential time‐stepping.

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Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Organization:: USDOE; Natural Sciences and Engineering Research Council of Canada (NSERC)

Grant/Contract Number:: AC52-07NA27344; RGPIN-2014-06032; RGPIN-2019-04155; RGPIN-2019-05692; DE‐AC52‐07NA27344

OSTI ID:: 1819027

Alternate ID(s):: OSTI ID: 1805044

Report Number(s):: LLNL-JRNL-789101; 987247

Journal Information:: Numerical Linear Algebra with Applications, Vol. 28, Issue 4; ISSN 1070-5325

Publisher:: WileyCopyright Statement

Country of Publication:: United States

Language:: English

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