In this paper, we present the new “asynchronous truncated multigrid-reduction-in-time” (AT-MGRIT) algorithm for introducing time parallelism to the solution of discretized time-dependent problems. The new algorithm is based on the multigrid-reduction-in-time (MGRIT) approach, which, in certain settings, is equivalent to another common multilevel parallel-in-time method, Parareal. In contrast to Parareal and MGRIT that both consider a global temporal grid over the entire time interval on the coarsest level, the AT-MGRIT algorithm uses truncated local time grids on the coarsest level, each grid covering certain temporal subintervals. These local grids can be solved completely in an independent way from each other, which reduces the sequential part of the algorithm and, thus, increases parallelism in the method. Here, we study the effect of using truncated local coarse grids on the convergence of the algorithm, both theoretically and numerically, and show, using challenging nonlinear problems, that the new algorithm consistently outperforms classical Parareal/MGRIT in terms of time to solution.


  1. parallel-in-time integration
  2. Parareal
  3. MGRIT
  4. truncated coarsest grids

MSC codes

  1. 65F10
  2. 65M22
  3. 65M55

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: S281 - S306
ISSN (online): 1095-7197


Submitted: 12 July 2021
Accepted: 26 July 2022
Published online: 22 November 2022


  1. parallel-in-time integration
  2. Parareal
  3. MGRIT
  4. truncated coarsest grids

MSC codes

  1. 65F10
  2. 65M22
  3. 65M55



Funding Information

Bundesministerium für Bildung und Forschung https://doi.org/10.13039/501100002347 : 05M18PXB
Los Alamos National Laboratory https://doi.org/10.13039/100008902 : LA-UR-21-26105

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