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

 

Operator splitting for the KdV equation


Authors: Helge Holden, Kenneth H. Karlsen, Nils Henrik Risebro and Terence Tao
Journal: Math. Comp. 80 (2011), 821-846
MSC (2010): Primary 35Q53; Secondary 65M12, 65M15
Published electronically: September 17, 2010
MathSciNet review: 2772097
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Abstract: We provide a new analytical approach to operator splitting for equations of the type $ u_t=Au+B(u)$, where $ A$ is a linear operator and $ B$ is quadratic. A particular example is the Korteweg-de Vries (KdV) equation $ u_t-u u_x+u_{xxx}=0$. We show that the Godunov and Strang splitting methods converge with the expected rates if the initial data are sufficiently regular.


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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

Kenneth H. Karlsen
Affiliation: Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO–0316 Oslo, Norway
Email: kennethk@math.uio.no

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

Terence Tao
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555
Email: tao@math.ucla.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2010-02402-0
PII: S 0025-5718(2010)02402-0
Keywords: KdV equation, operator splitting
Received by editor(s): June 6, 2009
Received by editor(s) in revised form: December 9, 2009
Published electronically: September 17, 2010
Additional Notes: Supported in part by the Research Council of Norway. This paper was written as part of the international research program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during the academic year 2008–09. The fourth author is supported by a grant from the MacArthur Foundation, the NSF Waterman award, and NSF grant DMS-0649473.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.