Portugaliae Mathematica
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Published online: 2013-07-05
Convergence of a finite difference method for the KdV and modified KdV equations with $L^2$ data
Paulo Amorim[1] and Mário Figueira[2] (1) Universidade de Lisboa, Portugal(2) Universidade de Lisboa, Portugal
We prove strong convergence of a semi-discrete finite difference method for the KdV and modified KdV equations. We extend existing results to non-smooth data (namely, in $L^2$), without size restrictions. Our approach uses a fourth order (in space) stabilization term and a special conservative discretization of the nonlinear term. Convergence follows from a smoothing effect and energy estimates. We illustrate our results with numerical experiments, including a numerical investigation of an open problem related to uniqueness posed by Y. Tsutsumi.
Keywords: Korteweg–de Vries equation, KdV equation, finite difference scheme
Amorim Paulo, Figueira Mário: Convergence of a finite difference method for the KdV and modified KdV equations with $L^2$ data. Port. Math. 70 (2013), 23-50. doi: 10.4171/PM/1924