Error estimates of finite difference schemes for the Korteweg-de Vries equation

This article deals with the numerical analysis of the Cauchy problem for the Korteweg-de Vries equation with a finite difference scheme. We consider the explicit Rusanov scheme for the hyperbolic flux term and a 4-point theta-scheme for the dispersive term. We prove the convergence under a hyperbolic Courant- Friedrichs-Lewy condition when theta >= 1/2 and under an 'Airy' Courant-Friedrichs-Lewy condition when theta >= 1/2. More precisely, we get a first-order convergence rate for strong solutions in the Sobolev space H-s (R), s >= 6 and extend this result to the nonsmooth case for initial data in H-s (R), with s >= 3/4, at the price of a reduction in the convergence order. Numerical simulations indicate that the orders of convergence may be optimal when s >= 3.