Abstract
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 $\theta \geq \frac{1}{2}$ and under an ‘Airy’ Courant–Friedrichs–Lewy condition when $\theta <\frac{1}{2}$. More precisely, we get a first-order convergence rate for strong solutions in the Sobolev space $H^s(\mathbb{R})$, $s \geq 6$ and extend this result to the nonsmooth case for initial data in $H^s(\mathbb{R})$, with $s\geq \frac{3}{4}$, at the price of a reduction in the convergence order. Numerical simulations indicate that the orders of convergence may be optimal when $s\geq 3$.
On the Cauchy problem for the modified Korteweg-de Vries equation with steplike finite-gap initial data