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
- Matematik LTH
- Partial differential equations
- Numerical Analysis
Publiceringsår
2013
Språk
Engelska
Sidor
1900-1910
Publikation/Tidskrift/Serie
SIAM Journal on Numerical Analysis
Volym
51
Issue
4
Fulltext
- Available as PDF - 355 kB
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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