Stability of linear multistep methods and applications to nonlinear parabolic problems.
type: Item
instance of: scholarly article
title: Stability of linear multistep methods and applications to nonlinear parabolic problems.
zbMATH Open document ID: 1041.65073
published in: Applied Numerical Mathematics
publication date: 2004-03-14T00:00:00Z
review text: Stability and convergence properties of linear multistep methods for nonlinear parabolic problems are investigated. This analysis admits variable stepsizes and is based on an abstract theory of sectorial operators and analytic semigroups in Banach spaces. This approach allows the stability factor to depend on the stepsize sequence and it must be guaranteed that this factor remains bounded by a reasonable constant. Firstly this approach was used for ordinary differential equations and further for linear parabolic problems. However, these resluts are not sufficient to obtain the convergence for nonlinear problems. So the aim of this paper is to derive the stability estimates and to develop a convergence theory of multistep methods for nonlinear problems. First stability resluts for arbitrary sectorial operators on compact time interval \( [0,T] \) are proved and then this theory is expanded to time dependent problems of type: \[ u^\prime(t) = A(t) u(t), 0<t\leq T, u(0) \text{ given }, \] with Hölder continuous operator \(A\). The obtained results are applied to semilinear parabolic problems that conclude for example reaction-diffusion or incompressible Navier-Stokes equations. Convergence results for finite times are shown. The last section is devoted to convergence resluts for fully nonlinear parabolic problems: \[ u^\prime(t) = F(t,u(t)), t>0 , u(0) \text{ given }. \] The long term behaviour of multistep discretization nearby an asymptotically stable equilibirum is studied and convergence results for smooth solutions on compact time intervals are proved.
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35K90
zbMATH DE Number: 2056712
zbMATH Keywords: linear multistep method
zbMATH Keywords: stability
zbMATH Keywords: variable stepsizes
zbMATH Keywords: semilinear parabolic problems
zbMATH Keywords: fully nonlinear parabolic problems
zbMATH Keywords: convergence
zbMATH Keywords: semidiscretization
zbMATH Keywords: sectorial operator
zbMATH Keywords: analytic semigroup
zbMATH Keywords: Navier-Stokes equation
zbMATH Keywords: Banach space
zbMATH Keywords: reaction-diffusion equation
author: Alexander Ostermann
author: Mechthild Thalhammer
author: Gabriela Kirlinger
reviewed by: Angela Handlovičová
MaRDI profile type: Publication
full work available at URL: j.apnum.2003.10.004
OpenAlex ID: W2131574086
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DOI: 10.1016/J.APNUM.2003.10.004
description: scientific article; zbMATH DE number 2056712