A finite difference scheme on graded meshes for time-fractional nonlinear Korteweg-de Vries equation

Abstract

In this paper, a finite difference scheme for time-fractional nonlinear Korteweg-de Vries (KdV) problems with the Caputo-type fractional derivative is presented. To deal with the weak singularity caused by the fractional derivative that the solution has in the initial layer, the well-known L1 scheme on graded meshes has been used for time discretization. Meanwhile, a nonlinear finite difference approximation on uniform meshes is proposed for spatial discretization. The existence, stability and convergence of the numerical solution are studied by the energy method. It is proved that the scheme is of min{2−α,rα} order convergence in time and of first-order convergence in space, where α is the order of fractional derivative and r is a parameter of graded meshes. Furthermore, in order to increase computational efficiency, the corresponding fast algorithm of the presented scheme has been considered. In numerical simulation, a valid approach based on linear interpolation is designed to test convergence rate for the scheme on temporal graded meshes. Numerical results demonstrate sharpness of the theoretical convergence estimate and effectiveness of the fast algorithm.