Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient

In this paper, we consider the numerical solutions of the time fractional sub-diffusion equation with the variable coefficient subject to both Dirichlet boundary conditions and Neumann boundary conditions. A compact difference scheme is proposed for solving the equation with Dirichlet boundary conditions. The unconditional stability and the global convergence of the scheme in the maximum norm are proved rigorously with the help of the newly introduced norms regarding to the variable coefficient. The convergence order is O(τ2-α+h4)O(τ2-α+h4), where ττ is the temporal grid size, αα is the order of fractional derivative and h is the spatial grid size. Besides, a box-type scheme is derived by introducing new intermediate variable for the problem with Neumann boundary conditions. The stability and the global convergence of box-type scheme in maximum norm are also presented. Numerical experiments are carried out to confirm the theoretical results of the proposed schemes.