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Parallel Implicit-Explicit General Linear Methods

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Abstract

High-order discretizations of partial differential equations (PDEs) necessitate high-order time integration schemes capable of handling both stiff and nonstiff operators in an efficient manner. Implicit-explicit (IMEX) integration based on general linear methods (GLMs) offers an attractive solution due to their high stage and method order, as well as excellent stability properties. The IMEX characteristic allows stiff terms to be treated implicitly and nonstiff terms to be efficiently integrated explicitly. This work develops two systematic approaches for the development of IMEX GLMs of arbitrary order with stages that can be solved in parallel. The first approach is based on diagonally implicit multi-stage integration methods (DIMSIMs) of types 3 and 4. The second is a parallel generalization of IMEX Euler and has the interesting feature that the linear stability is independent of the order of accuracy. Numerical experiments confirm the theoretical rates of convergence and reveal that the new schemes are more efficient than serial IMEX GLMs and IMEX Runge–Kutta methods.

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Acknowledgements

The authors acknowledge Advanced Research Computing at Virginia Tech for providing computational resources and technical support that have contributed to the results reported within this paper. URL: http://www.arc.vt.edu.

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Authors and Affiliations

  1. Computational Science Laboratory, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24060, USA

    Steven Roberts, Arash Sarshar & Adrian Sandu

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  1. Steven Roberts
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  2. Arash Sarshar
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Correspondence to Steven Roberts.

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On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

This work was funded by awards NSF CCF1613905, NSF ACI1709727, AFOSR DDDAS FA9550-17-1-0015, and by the Computational Science Laboratory at Virginia Tech.

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Roberts, S., Sarshar, A. & Sandu, A. Parallel Implicit-Explicit General Linear Methods. Commun. Appl. Math. Comput. 3, 649–669 (2021). https://doi.org/10.1007/s42967-020-00083-5

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  • DOI: https://doi.org/10.1007/s42967-020-00083-5

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