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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Operator splitting for partial differential equations with Burgers nonlinearity


Authors: Helge Holden, Christian Lubich and Nils Henrik Risebro
Journal: Math. Comp. 82 (2013), 173-185
MSC (2010): Primary 35Q53; Secondary 65M12, 65M15
Published electronically: June 12, 2012
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Abstract: We provide a new analytical approach to operator splitting for equations of the type $ u_t=Au+u u_x$ where $ A$ is a linear differential operator such that the equation is well-posed. Particular examples include the viscous Burgers equation, the Korteweg-de Vries (KdV) equation, the Benney-Lin equation, and the Kawahara equation. We show that the Strang splitting method converges with the expected rate if the initial data are sufficiently regular. In particular, for the KdV equation we obtain second-order convergence in $ H^r$ for initial data in $ H^{r+5}$ with arbitrary $ r\ge 1$.


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Additional Information

Helge Holden
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, NO–7491 Trondheim, Norway, — and — Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO–0316 Oslo, Norway
Email: holden@math.ntnu.no

Christian Lubich
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D–72076 Tübingen, Germany
Email: lubich@na.uni-tubingen.de

Nils Henrik Risebro
Affiliation: Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO–0316 Oslo, Norway
Email: nilshr@math.uio.no

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02624-X
PII: S 0025-5718(2012)02624-X
Keywords: Operator splitting, Burgers equation, KdV equation, Benney–Lin equation, Kawahara equation
Received by editor(s): February 8, 2011
Published electronically: June 12, 2012
Additional Notes: This work was supported in part by the Research Council of Norway.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.