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

Författare

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.

Avdelning/ar

Publiceringsår

2013

Språk

Engelska

Sidor

1900-1910

Publikation/Tidskrift/Serie

SIAM Journal on Numerical Analysis

Volym

51

Issue

4

Dokumenttyp

Artikel i tidskrift

Förlag

Society for Industrial and Applied Mathematics

Ämne

  • Mathematics

Nyckelord

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

Status

Published

Forskningsgrupp

  • Partial differential equations
  • Numerical Analysis

ISBN/ISSN/Övrigt

  • ISSN: 0036-1429