Convergence analysis of the Dirichlet-Neumann Waveform Relaxation algorithm for time fractional sub-diffusion and diffusion-wave equations in heterogeneous media

This article presents a comprehensive study on the convergence behavior of the Dirichlet-Neumann Waveform Relaxation algorithm applied to solve the time fractional sub-diffusion and diffusion-wave equations in multiple subdomains, considering the presence of some heterogeneous media. Our analysis focuses on estimating the convergence rate of the algorithm and investigates how this estimate varies with different fractional orders. Furthermore, we extend our analysis to encompass the 2D sub-diffusion case. To validate our findings, we conduct numerical experiments to verify the estimated convergence rate. The results confirm the theoretical estimates and provide empirical evidence for the algorithm’s efficiency and reliability. Moreover, we push the boundaries of the algorithm’s applicability by extending it to solve the time fractional Allen-Chan equation, a problem that exceeds our initial theoretical estimates. Remarkably, we observe that the algorithm performs exceptionally well in this extended scenario for both short and long-time windows.