Operator splitting for partial differential equations with Burgers nonlinearity

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, a...

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Bibliographic Details
Published in:Mathematics of computation 2013-01, Vol.82 (281), p.173-185
Main Authors: HOLDEN, HELGE, LUBICH, CHRISTIAN, RISEBRO, NILS HENRIK
Format: Article
Language:English
Subjects:
Approximation
Burger equation
Cauchy problem
Commutators
Error bounds
Mathematical problems
Nonlinearity
Perceptron convergence procedure
Sobolev spaces
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Summary: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.
ISSN:0025-5718
1088-6842
DOI:10.1090/S0025-5718-2012-02624-X