A Convergence Analysis of the Peaceman-Rachford Scheme for Semilinear Evolution Equations
(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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/3972343
- author
- Hansen, Eskil LU and Henningsson, Erik LU
- organization
- publishing date
- 2013
- type
- Contribution to journal
- 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
- 2024-01-06 11:39:47
@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}}, }