Convergence of a fully discrete finite difference scheme for the Korteweg-de Vries equation
This provides rigorous numerical analysis for a key nonlinear PDE, benefiting researchers in computational PDEs.
The paper proves convergence of a fully discrete finite difference scheme for the Korteweg-de Vries equation, showing convergence to classical solutions for high-regularity initial data and to weak solutions for low-regularity data.
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 $u|_{t=0}=u_0$ is of high regularity, $u_0\in H^3(\R)$, the scheme is shown to converge to a classical solution, and if the regularity of the initial data is smaller, $u_0\in L^2(\R)$, then the scheme converges strongly in $L^2(0,T;L^2_{\mathrm{loc}}(\R))$ to a weak solution.