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(NM log(M)) operations, where M is the number of time steps and N is the number of spatial grid points. A semialgebraic mode analysis is also developed to theoretically confirm the good results obtained. Several numerical experiments, including examples with nonsmooth solutions and a nonlinear problem with applications in porous media, are presented.

Additional Metadata
Keywords Multigrid waveform relaxation, Semialgebraic mode analysis, Time-fractional heat equation
Persistent URL dx.doi.org/10.1137/16M1090193
Journal SIAM Journal on Scientific Computing
Grant This work was funded by the European Commission 7th Framework Programme; grant id h2020/705402 - Efficient numerical methods for deformable porous media. Application to carbon dioxide storage. (poro sos)
Citation
Gaspar, F.J, & Rodrigo, C. (2017). Multigrid waveform relaxation for the time-fractional heat equation. SIAM Journal on Scientific Computing, 39(4), A1201–A1224. doi:10.1137/16M1090193