Dirichlet-Neumann Waveform Relaxation Algorithm for Time Fractional Diffusion Equation in Heterogeneous Media

TL;DR

This paper analyzes the convergence of Dirichlet-Neumann waveform relaxation algorithms for time-fractional diffusion equations in heterogeneous media, identifying optimal parameters and their dependence on fractional order and domain dimensions.

Contribution

It provides a convergence analysis and optimal relaxation parameters for waveform relaxation algorithms applied to time-fractional diffusion equations in heterogeneous media.

Findings

Optimal relaxation parameters are identified through numerical experiments.

Convergence rate depends on fractional order and time.

Analysis extends to both 1D and 2D cases.

Abstract

In this article, we have studied the convergence behavior of the Dirichlet-Neumann waveform relaxation algorithms for time-fractional sub-diffusion and diffusion wave equations in 1D \& 2D for regular domains, where the dimensionless diffusion coefficient takes different constant values in different subdomains. From numerical experiments, we first capture the optimal relaxation parameters. Using these optimal relaxation parameters, our analysis estimates the rate of change of the convergence behavior against the fractional order and time. We have performed our analysis in multiple subdomain cases for both in 1D \& 2D.