Convergence of a fully discrete finite difference scheme for the Korteweg-de Vries equation

Abstract

We prove convergence of a fully discrete finite difference scheme for the Korteweg–de Vries equation. Both the decaying case on the full line and the periodic case are considered. If the initial data u|t=0 = u0 is of high regularity, u0 ∈ H3 (R), the scheme is shown to converge to a classical solution, and if the regularity of the initial data is smaller, u0 ∈ L2 (R), then the scheme converges strongly in L2 (0, T; L2 loc(R)) to a weak solution.