Convergence of multistep time discretizations of nonlinear dissipative evolution equations
Hansen, Eskil LU
(2006)
In SIAM Journal on Numerical Analysis
44(1).
p.55-65
- Abstract
- Global error bounds are derived for multistep time discretizations of fully nonlinear evolution equations on infinite dimensional spaces. In contrast to earlier studies, the analysis presented here is not based on linearization procedures but on the fully nonlinear framework of logarithmic Lipschitz constants and nonlinear semigroups. The error bounds reveal how the contractive or dissipative behavior of the vector field, governing the evolution, and the properties of the multistep method influence the convergence. A multistep method which is consistent of order p is proven to be convergent of the same order when the vector field is contractive or strictly dissipative, i.e., of the same order as in the ODE-setting. In the contractive... (More)
- Global error bounds are derived for multistep time discretizations of fully nonlinear evolution equations on infinite dimensional spaces. In contrast to earlier studies, the analysis presented here is not based on linearization procedures but on the fully nonlinear framework of logarithmic Lipschitz constants and nonlinear semigroups. The error bounds reveal how the contractive or dissipative behavior of the vector field, governing the evolution, and the properties of the multistep method influence the convergence. A multistep method which is consistent of order p is proven to be convergent of the same order when the vector field is contractive or strictly dissipative, i.e., of the same order as in the ODE-setting. In the contractive context it is sufficient to require strong zero-stability of the method, whereas strong A-stability is sufficient in the dissipative case. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/415285
- author
- Hansen, Eskil
LU
- organization
- publishing date
- 2006
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- convergence, stability, multistep methods, dissipative maps, nonlinear evolution equations, logarithmic Lipschitz constants
- in
- SIAM Journal on Numerical Analysis
- volume
- 44
- issue
- 1
- pages
- 55 - 65
- publisher
- Society for Industrial and Applied Mathematics
- external identifiers
-
- wos:000236099800004
- scopus:33748995135
- ISSN
- 0036-1429
- DOI
- 10.1137/040610362
- 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
- 884eb294-9ca1-42fb-9a49-a9c00056c8b2 (old id 415285)
- date added to LUP
- 2016-04-01 17:11:18
- date last changed
- 2024-02-27 13:32:39
@article{884eb294-9ca1-42fb-9a49-a9c00056c8b2,
abstract = {{Global error bounds are derived for multistep time discretizations of fully nonlinear evolution equations on infinite dimensional spaces. In contrast to earlier studies, the analysis presented here is not based on linearization procedures but on the fully nonlinear framework of logarithmic Lipschitz constants and nonlinear semigroups. The error bounds reveal how the contractive or dissipative behavior of the vector field, governing the evolution, and the properties of the multistep method influence the convergence. A multistep method which is consistent of order p is proven to be convergent of the same order when the vector field is contractive or strictly dissipative, i.e., of the same order as in the ODE-setting. In the contractive context it is sufficient to require strong zero-stability of the method, whereas strong A-stability is sufficient in the dissipative case.}},
author = {{Hansen, Eskil}},
issn = {{0036-1429}},
keywords = {{convergence; stability; multistep methods; dissipative maps; nonlinear evolution equations; logarithmic Lipschitz constants}},
language = {{eng}},
number = {{1}},
pages = {{55--65}},
publisher = {{Society for Industrial and Applied Mathematics}},
series = {{SIAM Journal on Numerical Analysis}},
title = {{Convergence of multistep time discretizations of nonlinear dissipative evolution equations}},
url = {{http://dx.doi.org/10.1137/040610362}},
doi = {{10.1137/040610362}},
volume = {{44}},
year = {{2006}},
}