An α-robust finite difference method for a time-fractional radially symmetric diffusion problem

A time-fractional diffusion problem is considered on a spherically symmetric domain in R-d for d = 1,2,3. The solution of such a problem is shown in general to have a weak singularity near the initial time t = 0, and bounds on certain derivatives of this solution are derived. To solve the problem numerically, spatial polar coordinates are used; a finite difference method is constructed on a mesh that is graded in time and spherical in space. The discretisation uses the L1 scheme in time and a polar-coordinate discretisation of the diffusion term. Its convergence is analysed and error bounds are derived that are robust in alpha, the order of the time derivative, as alpha -> 1(-). Numerical experiments show that our results are sharp.