Multigrid waveform relaxation for the time-fractional heat equation

TL;DR

This paper introduces a multigrid waveform relaxation method for efficiently solving the time-fractional heat equation, leveraging Toeplitz structure and parallel-in-time algorithms to handle nonlocal operators.

Contribution

It develops a novel parallel-in-time multigrid algorithm tailored for the nonlocal fractional heat equation, with theoretical analysis and practical efficiency improvements.

Findings

Computational cost is $O(N M \,\log(M))$ operations.

Method performs well with non-smooth solutions.

Effective for nonlinear porous media problems.

Abstract

In this work, we propose an efficient and robust multigrid method for solving the time-fractional heat equation. Due to the nonlocal property of fractional differential operators, numerical methods usually generate systems of equations for which the coefficient matrix is dense. Therefore, the design of efficient solvers for the numerical simulation of these problems is a difficult task. We develop a parallel-in-time multigrid algorithm based on the waveform relaxation approach, whose application to time-fractional problems seems very natural due to the fact that the fractional derivative at each spatial point depends on the values of the function at this point at all earlier times. Exploiting the Toeplitz-like structure of the coefficient matrix, the proposed multigrid waveform relaxation method has a computational cost of $O(N M \log(M))$ operations, where $M$ is the number of time steps and $N$ is the number of spatial grid points. A semi-algebraic mode analysis is also developed to theoretically confirm the good results obtained. Several numerical experiments, including examples with non-smooth solutions and a nonlinear problem with applications in porous media, are presented.