A compact ADI scheme for two-dimensional fractional sub-diffusion equation with Neumann boundary condition

摘要

In this paper, we develop an effective numerical method for solving the fractional sub-diffusion equation with Neumann boundary conditions. The time fractional derivative is approximated by the L1 scheme on graded meshes, the spatial discretization is done by using the compact finite difference methods. By adding some corrected terms, the fully discrete alternating direction implicit (ADI) method is obtained. Convergence of the scheme is obtained under the assumptions of the weak singularity of solutions. The extension of numerical scheme to the three-dimensional case is presented. Finally, the effectiveness of the proposed method is confirmed by several numerical experiments.