Finite difference methods for the time fractional diffusion equation on non-uniform meshes

Since fractional derivatives are integrals with weakly singular kernel, the discretization on the uniform mesh may lead to poor accuracy. The finite difference approximation of Caputo derivative on non-uniform meshes is investigated in this paper. The method is applied to solve the fractional diffusion equation and a semi-discrete scheme is obtained. The unconditional stability and H1H1 norm convergence are proved. A fully discrete difference scheme is constructed with space discretization by compact difference method. The error estimates are established for two kinds of nonuniform meshes. Numerical tests are carried out to support the theoretical results and comparing with the method on uniform grid shows the efficiency of our methods. Moreover, a moving local refinement technique is introduced to improve the temporal accuracy of numerical solution.