Scientific article
Open access

Optimization of Schwarz waveform relaxation over short time windows

Published inNumerical Algorithms, vol. 64, no. 2, p. 221-243
  • Open Access - Licence nationale Springer
Publication date2013

Schwarz waveform relaxation algorithms (SWR) are naturally parallel solvers for evolution partial differential equations. They are based on a decomposition of the spatial domain into subdomains, and a partition of the time interval of interest into time windows. On each time window, an iteration, during which subproblems are solved in space-time subdomains, is then used to obtain better and better approximations of the overall solution. The information exchange between subdomains in space-time is performed through classical or optimized transmission conditions (TCs). We analyze in this paper the optimization problem when the time windows are short. We use as our model problem the optimized SWR algorithm with Robin TCs applied to the heat equation. After a general convergence analysis using energy estimates, we prove that in one spatial dimension, the optimized Robin parameter scales like the inverse of the length of the time window, which is fundamentally different from the known scaling on general bounded time windows, which is like the inverse of the square root of the time window length. We illustrate our analysis with a numerical experiment.

  • Schwarz waveform relaxation
  • Optimized Robin transmission conditions
  • Short time windows
  • Heat equation
Citation (ISO format)
COURVOISIER, Yves, GANDER, Martin Jakob. Optimization of Schwarz waveform relaxation over short time windows. In: Numerical Algorithms, 2013, vol. 64, n° 2, p. 221–243. doi: 10.1007/s11075-012-9662-y
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Article (Published version)
ISSN of the journal1017-1398

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