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High-order splitting schemes for semilinear evolution equations

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Abstract

We first derive necessary and sufficient stiff order conditions, up to order four, for exponential splitting schemes applied to semilinear evolution equations. The main idea is to identify the local splitting error as a sum of quadrature errors. The order conditions of the quadrature rules then yield the stiff order conditions in an explicit fashion, similarly to that of Runge–Kutta schemes. Furthermore, the derived stiff conditions coincide with the classical non-stiff conditions. Secondly, we propose an abstract convergence analysis, where the linear part of the vector field is assumed to generate a group or a semigroup and the nonlinear part is assumed to be smooth and to satisfy a set of compatibility requirements. Concrete applications include nonlinear wave equations and diffusion-reaction processes. The convergence analysis also extends to the case where the nonlinear flows in the exponential splitting scheme are approximated by a sufficiently accurate one-step method.

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Authors and Affiliations

  1. Centre for Mathematical Sciences, Lund University, P.O. Box 118, 221 00, Lund, Sweden

    Eskil Hansen

  2. Institut für Mathematik, Universität Innsbruck, Technikerstraße 13, 6020, Innsbruck, Austria

    Alexander Ostermann

Authors
  1. Eskil Hansen
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  2. Alexander Ostermann
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Correspondence to Eskil Hansen.

Additional information

Communicated by Christian Lubich.

The work of E. Hansen was supported by the Swedish Research Council under Grant 621-2011-5588.

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Hansen, E., Ostermann, A. High-order splitting schemes for semilinear evolution equations. Bit Numer Math 56, 1303–1316 (2016). https://doi.org/10.1007/s10543-016-0604-2

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  • DOI: https://doi.org/10.1007/s10543-016-0604-2

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