Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions

An effective finite difference scheme is considered for solving the time fractional sub-diffusion equation with Neumann boundary conditions. A difference scheme combining the compact difference approach the spatial discretization and L1L1 approximation for the Caputo fractional derivative is proposed and analyzed. Although the spatial approximation order at the Neumann boundary is one order lower than that for interior mesh points, the unconditional stability and the global convergence order O(τ2-α+h4)O(τ2-α+h4) in discrete L2L2 norm of the compact difference scheme are proved rigorously, where ττ is the temporal grid size and h is the spatial grid size. Numerical experiments are included to support the theoretical results, and comparison with the related works are presented to show the effectiveness of our method.