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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)
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author
organization
publishing date
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
2021-01-06 05:05:19
@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},
  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},
}