Waveform relaxation for fractional sub-diffusion equations

We report a new kind of waveform relaxation (WR) method for general semi-linear fractional sub-diffusion equations, and analyze the upper bound for the iteration errors. It indicates that the WR method is convergent superlinearly, and the convergence rate is dependent on the order of the time-fractional derivative and the length of the time interval. In order to accelerate the convergence, we present the windowing WR method. Then, we elaborate the parallelism based on the discrete windowing WR method, and present the corresponding fast evaluation formula. Numerical experiments are carried out to verify the effectiveness of the theoretic work.