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Multirate generalized additive Runge Kutta methods

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Abstract

This work constructs a new class of multirate schemes based on the recently developed generalized additive Runge–Kutta (GARK) methods (Sandu and Günther, SIAM J Numer Anal, 53(1):17–42, 2015). Multirate schemes use different step sizes for different components and for different partitions of the right-hand side based on the local activity levels. We show that the new multirate GARK family includes many well-known multirate schemes as special cases. The order conditions theory follows directly from the GARK accuracy theory. Nonlinear stability and monotonicity investigations show that these properties are inherited from the base schemes provided that additional coupling conditions hold.

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Acknowledgments

The work of A. Sandu has been supported in part by NSF through awards NSF OCI-8670904397, NSF CCF-0916493, NSF DMS-0915047, NSF CMMI-1130667, NSF CCF-1218454, AFOSR FA9550-12-1-0293-DEF, AFOSR 12-2640-06, and by the Computational Science Laboratory at Virginia Tech. The work of M. Günther has been supported in part by BMBF through grant 03MS648E.

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Authors and Affiliations

  1. Institute of Mathematical Modelling, Analysis and Compuational Mathematics (IMACM), Bergische Universität Wuppertal, Gaußstrasse 20, 42119, Wuppertal, Germany

    Michael Günther

  2. Computational Science Laboratory, Department of Computer Science, Virginia Polytechnic Institute and State University, 2202 Kraft Drive, Blacksburg, VA, 24060, USA

    Adrian Sandu

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  1. Michael Günther
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  2. Adrian Sandu
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Correspondence to Michael Günther.

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Günther, M., Sandu, A. Multirate generalized additive Runge Kutta methods. Numer. Math. 133, 497–524 (2016). https://doi.org/10.1007/s00211-015-0756-z

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  • DOI: https://doi.org/10.1007/s00211-015-0756-z

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