A Note on the Convergence of a Crank-Nicolson Scheme for the KdV Equation

Abstract

The aim of this paper is to establish the convergence of a fully discrete Crank-Nicolson type Galerkin scheme for the Cauchy problem associated to the KdV equation. The convergence is achieved for initial data in $L^2$, and we show that the scheme converges strongly in $L^2(0, T; L^2_{loc}(\mathbb{R}))$ to a weak solution for some $T >0$. Finally, the convergence is illustrated by a numerical example.