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A second-order Magnus-type integrator for quasi-linear parabolic problems


Authors: C. González and M. Thalhammer
Journal: Math. Comp. 76 (2007), 205-231
MSC (2000): Primary 35K55, 35K90, 65L20, 65M12
DOI: https://doi.org/10.1090/S0025-5718-06-01883-7
Published electronically: August 15, 2006
MathSciNet review: 2261018
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Abstract: In this paper, we consider an explicit exponential method of classical order two for the time discretisation of quasi-linear parabolic problems. The numerical scheme is based on a Magnus integrator and requires the evaluation of two exponentials per step. Our convergence analysis includes parabolic partial differential equations under a Dirichlet boundary condition and provides error estimates in Sobolev spaces. In an abstract formulation the initial boundary value problem is written as an initial value problem on a Banach space $ X$

$\displaystyle u'(t) = A\big(u(t)\big) u(t), \quad 0 < t \leq T, \qquad u(0)$    given$\displaystyle , $

involving the sectorial operator $ A(v):D \to X$ with domain $ D \subset X$ independent of $ v \in V \subset X$. Under reasonable regularity requirements on the problem, we prove the stability of the numerical method and derive error estimates in the norm of certain intermediate spaces between $ X$ and $ D$. Various applications and a numerical experiment illustrate the theoretical results.


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Additional Information

C. González
Affiliation: Departamento de Matemática Aplicada y Computación, Facultad de Ciencias, Universidad de Valladolid, E-47011 Valladolid, Spain
Email: cesareo@mac.cie.uva.es

M. Thalhammer
Affiliation: Institut für Mathematik, Fakultät für Mathematik, Informatik und Physik, Universität Innsbruck, Technikerstrasse 25/7, A-6020 Innsbruck, Austria
Email: Mechthild.Thalhammer@uibk.ac.at

DOI: https://doi.org/10.1090/S0025-5718-06-01883-7
Keywords: Quasi-linear parabolic problems, Magnus integrators, stability, convergence
Received by editor(s): December 20, 2004
Received by editor(s) in revised form: September 30, 2005
Published electronically: August 15, 2006
Article copyright: © Copyright 2006 American Mathematical Society