L1-robust analysis of a fourth-order block-centered finite difference method for two-dimensional variable-coefficient time-fractional reaction-diffusion equations

In this paper, we develop a high-order finite difference scheme for the two-dimensional time-fractional reaction-diffusion equation with variably diffusion coefficient, in which the non-uniform L1 time stepping method on graded mesh is utilized for temporal discretization to compensate for the possible temporal accuracy lost caused by the initial weak singularity, and by introducing a flux variable, a fourth-order block-centered finite difference (BCFD) method on uniform staggered grids is developed for spatial discretization. By utilizing the well-defined discrete complementary convolution kernels and alpha-robust fractional Gronwall inequalities, the alpha- robust unconditional stability and optimal-order error estimate for the primal variable are rigorously proved. Meanwhile, the alpha-robust stability and fourth-order error estimate for the flux variable under a rough time stepsize condition are also proved. To further reduce the memory requirement and computational cost, a fast sum-of-exponentials (SOE)-based L1-BCFD algorithm is also given. Finally, numerical examples are tested to show the efficiency and accuracy of the developed methods.