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A Convergence Analysis of the Peaceman-Rachford Scheme for Semilinear Evolution Equations

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Author

Summary, in English

The Peaceman--Rachford scheme is a commonly used splitting method for discretizing semilinear evolution equations, where the vector fields are given by the sum of one linear and one nonlinear dissipative operator. Typical examples of such equations are reaction-diffusion systems and the damped wave equation. In this paper we conduct a convergence analysis for the Peaceman--Rachford scheme in the setting of dissipative evolution equations on Hilbert spaces. We do not assume Lipschitz continuity of the nonlinearity, as previously done in the literature. First or second order convergence is derived, depending on the regularity of the solution, and a shortened proof for $o(1)$-convergence is given when only a mild solution exits. The analysis is also extended to the Lie scheme in a Banach space framework. The convergence results are illustrated by numerical experiments for Caginalp's solidification model and the Gray--Scott pattern formation problem.

Department/s

Publishing year

2013

Language

English

Pages

1900-1910

Publication/Series

SIAM Journal on Numerical Analysis

Volume

51

Issue

4

Document type

Journal article

Publisher

Society for Industrial and Applied Mathematics

Topic

  • Mathematics

Keywords

  • Peaceman--Rachford scheme
  • convergence order
  • semilinear evolution equations
  • reaction-diusion systems
  • dissipative operators

Status

Published

Research group

  • Partial differential equations
  • Numerical Analysis

ISBN/ISSN/Other

  • ISSN: 0036-1429