Optimized Overlapping Schwarz Waveform Relaxation for a Class of Time-Fractional Diffusion Problems

Abstract

For parabolic PDEs with integer-order temporal derivative, if we use the Schwarz waveform relaxation (SWR) algorithm with Robin transmission conditions as the solver, the so-called equioscillation principle is an important concept to get a good Robin parameter, which has a significant effect on the convergence rate of the algorithm. Surprisingly, as we show in this paper such a principle may result in rather disappointing Robin parameter for the SWR algorithm when we use it to solve time-fractional PDEs. For a class of time-fractional diffusion equations, by analyzing a new min–max problem we get much better Robin parameter, which is found very close to the best one that we can make through numerical optimizations and numerical experiments. To use the SWR algorithm in practice, we apply the kernel reduction technique proposed recently by Baffet and Hesthaven to treat the convolutions with kernel function of the form \(\mathscr {K}_{\gamma }(t)=t^{-\gamma }/\varGamma (1-\gamma )\), where \(\gamma \in (0, 1)\). For time-fractional PDEs with this kind of kernel function, the kernel reduction technique results in efficient one-step numerical schemes. Numerical results obtained by using this technique confirm our theoretical conclusions very well.