On some high-order accurate fully discrete Galerkin methods for the Korteweg-de Vries equation

Abstract:

We construct and analyze fully discrete Galerkin (finite-element) methods of high order of accuracy for the numerical solution of the periodic initial-value problem for the Korteweg-de Vries equation. The methods are based on a standard space discretization using smooth periodic splines on a uniform mesh. For the time stepping, we use two schemes of third (resp. fourth) order of accuracy which are modifications of well-known, diagonally implicit Runge-Kutta methods and require the solution of two (resp. three) nonlinear systems of equations at each time step. These systems are solved approximately by Newton’s method. Provided the initial iterates are chosen in a specific, accurate way, we show that only one Newton iteration per system is needed to preserve the stability and order of accuracy of the scheme. Under certain mild restrictions on the space mesh length and the time step we prove ${L^2}$-error estimates of optimal rate of convergence for both schemes.