Convergence of a finite difference method for the KdV and modified KdV equations with \(L^2\) data (Q360093)

The paper deals with the numerical solution of the Korteweg-de Vries (KdV) and the modified KdV equations with the aid of the finite difference method. The strong convergence is proved even for non-smooth data (namely, in \(L^2\)), without size restrictions. The approach is based on 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. Numerical experiments showing the order of convergence are presented. Moreover, a numerical investigation of an open problem related to uniqueness posed by \textit{Y. Tsutsumi} [SIAM J. Math. Anal. 20, No. 3, 582--588 (1989; Zbl 0679.35078)] is given.