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Error analysis of operator splitting for the Korteweg - de Vries equation

Seminar
Wednesday, June 4, 2014 at 15:00 – GSSI, Room D

Prof. Christian Lubich

 

ABSTRACT

This talk is based on joint work with Helge Holden and Nils Henrik Risebro. We provide an error analysis of operator splitting for the Korteweg - de Vries (KdV) equation. It is shown show that the Strang splitting method converges with the expected rate if the initial data are sufficiently regular: we obtain second-order convergence in $H^r$ for initial data in $H^{r+5}$ with arbitrary $r\ge 1$. This result improves that of Holden, Karlsen, Risebro and Tao (2011), where second-order convergence is shown in $H^r$ for initial data in $H^{r+9}$ with $r\ge 8$.
The proof derives local error estimates, for the error after one step with starting value in $H^{r+5}$, by interpreting the principal error terms as quadrature errors. Using the local error estimates and regularity results for the KdV and Burgers equations, boundedness of the numerical solution in $H^{r+5}$ is shown over arbitrary finite time intervals. These estimates are combined, via Lady Windermere's fan with error propagation by the exact solutions of the KdV equation, to finally obtain the result.