Convergence of a Higher Order Scheme for the Korteweg--De Vries Equation

Abstract

We study the convergence of higher order schemes for the Cauchy problem associated with the KdV equation. More precisely, we design a Galerkin-type implicit scheme which has higher order accuracy in space and first order accuracy in time. The convergence is established 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. Finally, the convergence is illustrated by several numerical examples.