Parareal for Diffusion Problems with Space- and Time-Dependent Coefficients
The very rapidly increasing number of cores in state-of-the-art supercomputers fuels both the need for and the interest in novel numerical algorithms inherently designed to feature concurrency. In addition to the mature field of space-parallel approaches (e.g. domain decomposition techniques), time-parallel methods that allow concurrency along the temporal dimension are now an increasingly active field of research, although first ideas, like in , go back several decades. A prominent and widely studied algorithm in this area is Parareal, introduced in , which has the advantage that one can couple and reuse classical time-stepping schemes in an iterative fashion to parallelize in time. However, there also exist a number of other approaches, e.g. the “parallel implicit time algorithm” (PITA) from , the “parallel full approximation scheme in space and time” (PFASST) from  or “revisionist integral deferred corrections” (RIDC) from  to name a few. Parareal in particular and temporal parallelism in general has been considered early as an addition to spatial parallelism in order to extend strong scaling limits, see . Efficacy of this approach in large-scale parallel simulations on hundreds of thousands of cores has been demonstrated for the PFASST algorithm in .
KeywordsParareal Iteration Iteration Matrix Fixed Point Iteration Implicit Euler Method Domain Decomposition Technique
This work was supported by the Swiss National Science Foundation (SNSF) under the lead agency agreement through the project “ExaSolvers” within the Priority Programme 1648 “Software for Exascale Computing” (SPPEXA) of the Deutsche Forschungsgemeinschaft (DFG). The authors thankfully acknowledge support from Achim Schädle, who provided parts of the used code.
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