Error estimates of finite difference schemes for the Korteweg-de Vries equation
For researchers in numerical analysis of dispersive PDEs, this provides rigorous convergence guarantees for a specific finite difference scheme, though the result is incremental.
The authors prove convergence of a finite difference scheme for the Korteweg-de Vries equation under Courant-Friedrichs-Lewy conditions, achieving first-order convergence for strong solutions and extending to non-smooth initial data with a loss in convergence order.
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 Rusanov scheme for the hyperbolic flux term and a 4-points $θ$-scheme for the dispersive term. We prove the convergence under a hyperbolic Courant-Friedrichs-Lewy condition when $θ\geq \frac{1}{2}$ and under an "Airy" Courant-Friedrichs-Lewy condition when $θ<\frac{1}{2}$. More precisely, we get the first order convergence rate for strong solutions in the Sobolev space $H^s(\mathbb{R})$, $s \geq 6$ and extend this result to the non-smooth case for initial data in $H^s(\mathbb{R})$, with $s\geq \frac{3}{4}$ , to the price of a loss in the convergence order. Numerical simulations indicate that the orders of convergence may be optimal when $s\geq3$.