Local error estimates of the fourth-order compact difference scheme for a time-fractional diffusion-wave equation

Abstract

This paper considers the numerical approximation for the time-fractional diffusion-wave equation with the order α∈(1,2), whose solution behaves a weak singularity at t=0. To construct the high-order scheme, the intermediate variable w=Dtα/2(u−tϕ˜) is introduced, then we can rewrite the original problem as a coupled system, where u is the solution of the time-fractional diffusion-wave equation and ϕ˜=ut(x,0). By using the L1 scheme on graded meshes in temporal direction and the compact difference scheme in spatial direction, a fully discrete scheme is constructed for the coupled system. Furthermore, the stability and the local error estimate of the proposed scheme is given. Finally, numerical examples are presented to verify the accuracy and efficiency of the proposed method.