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

Hansen, Eskil LU and Henningsson, Erik LU (2013) In SIAM Journal on Numerical Analysis 51(4). p.1900-1910
Abstract
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... (More)
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. (Less)
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publication status
published
subject
keywords
Peaceman--Rachford scheme, convergence order, semilinear evolution equations, reaction-diusion systems, dissipative operators
in
SIAM Journal on Numerical Analysis
volume
51
issue
4
pages
11 pages
publisher
Society for Industrial and Applied Mathematics
external identifiers
  • wos:000323892000002
  • scopus:84888858490
ISSN
0036-1429
DOI
10.1137/120890570
language
English
LU publication?
yes
additional info
The information about affiliations in this record was updated in December 2015. The record was previously connected to the following departments: Numerical Analysis (011015004)
id
90e23b0c-f178-4005-bd02-fb7b6ae361dc (old id 3972343)
date added to LUP
2016-04-01 10:14:47
date last changed
2022-04-04 03:53:16
@article{90e23b0c-f178-4005-bd02-fb7b6ae361dc,
  abstract     = {{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.}},
  author       = {{Hansen, Eskil and Henningsson, Erik}},
  issn         = {{0036-1429}},
  keywords     = {{Peaceman--Rachford scheme; convergence order; semilinear evolution equations; reaction-diusion systems; dissipative operators}},
  language     = {{eng}},
  number       = {{4}},
  pages        = {{1900--1910}},
  publisher    = {{Society for Industrial and Applied Mathematics}},
  series       = {{SIAM Journal on Numerical Analysis}},
  title        = {{A Convergence Analysis of the Peaceman-Rachford Scheme for Semilinear Evolution Equations}},
  url          = {{https://lup.lub.lu.se/search/files/1686524/3972352}},
  doi          = {{10.1137/120890570}},
  volume       = {{51}},
  year         = {{2013}},
}