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Accurate Implicit–Explicit General Linear Methods with Inherent Runge–Kutta Stability

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Abstract

We investigate implicit–explicit (IMEX) general linear methods (GLMs) with inherent Runge–Kutta stability (IRKS) for differential systems with non-stiff and stiff processes. The construction of such formulas starts with implicit GLMs with IRKS which are A- and L-stable, and then we ‘remove’ implicitness in non-stiff terms by extrapolating unknown stage derivatives by stage derivatives which are already computed by the method. Then we search for IMEX schemes with large regions of absolute stability of the ‘explicit part’ of the method assuming that the ‘implicit part’ of the scheme is \(A(\alpha )\)-stable for some \(\alpha \in (0,\pi /2]\). Examples of highly stable IMEX GLMs are provided of order \(1\le p\le 4\). Numerical examples are also given which illustrate good performance of these schemes.

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Acknowledgments

The research reported in this paper was started during the visit of the first author (MB) to the Arizona State University in November 2014. This author wish to express his gratitude to the School of Mathematical and Statistical Sciences for hospitality during this visit.

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Authors and Affiliations

  1. Faculty of Applied Mathematics, AGH University of Science and Technology, Kraków, Poland

    Michał Braś & Zdzisław Jackiewicz

  2. Dipartimento di Matematica e Applicazioni, Università di Napoli-“Federico II”, 80126, Naples, Italy

    Giuseppe Izzo

  3. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287, USA

    Zdzisław Jackiewicz

Authors
  1. Michał Braś
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  2. Giuseppe Izzo
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  3. Zdzisław Jackiewicz
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Correspondence to Giuseppe Izzo.

Additional information

The work of Michał Braś was supported by the National Science Center under Grant DEC-2011/01/N/ST1/02672 and the Polish Ministry of Science and Higher Education.

The work of Giuseppe Izzo was partially supported by GNCS-INdAM.

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Braś, M., Izzo, G. & Jackiewicz, Z. Accurate Implicit–Explicit General Linear Methods with Inherent Runge–Kutta Stability. J Sci Comput 70, 1105–1143 (2017). https://doi.org/10.1007/s10915-016-0273-y

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