Optimal order finite difference/local discontinuous Galerkin method for variable-order time-fractional diffusion equation

Abstract

In this paper, an accurate numerical method is presented to solve a class of variable-order fractional diffusion problem. The problem first is discretized by a finite difference method in temporal direction, and then a local discontinuous Galerkin method in space. The stability and L2 convergence of the proposed scheme are derived for all variable-order α(t)∈(0,1). We prove that the scheme is of accuracy-order O(τ+hk+1), where τ, h and k are temporal step sizes, spatial step sizes and the degree of piecewise Pk polynomials, respectively. Some numerical experiments are provided to verify the theoretical analysis and high-accuracy of the proposed method.