A Compact Difference Scheme for Time Fractional Diffusion Equation with Neumann Boundary Conditions

Abstract

This paper is devoted to the numerical treatment of time fractional diffusion equation with Neumann boundary conditions. A compact difference scheme is derived for solving this problem, by combining the classic finite difference method for Caputo derivative in time, the second order central difference method in space and the compact difference treatment for Neumann boundary conditions. The solvability, stability and convergence of this scheme are rigorously discussed. We prove that the convergence order of this proposed scheme is O(τ2 − α + h2), where τ, α and h are the time step size, the index of fractional derivative and space step size respectively. Numerical experiments are carried out to demonstrate the theoretical analysis.