European Union’s Horizon 2020 research and innovation programme

Marie Sklodowska-Curie grant agreement NO 705402, POROSOS

FAST SOLVERS FOR THE TIME FRACTIONAL HEAT EQUATION

We have developed a 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. In the case of uniform grids in time, we have proposed a multigrid waveform relaxation method with 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. Here, we exploit the Toeplitz-like structure of the coefficient matrix.

To maintain an optimal complexity in the case of non-uniform grids in time, we have approximated the coefficient matrix corresponding to the temporal discretization ny its hierarchical matrix (H-matrix) representation. The proposed method has a computational cost of O(k N M log(M)) where again M is the number of time steps, N is the number of spatial grid points and k is a parameter which controls the accuracy of the H-matrix approximation.

PUBLICATIONS:

Multigrid waveform relaxation for the time-fractional heat equation. F.J. Gaspar, C. Rodrigo. SIAM Journal on Scientific Computing 39. (2017) : LINK

Using hierarchical matrices in the solution of the time-fractional heat equation by multigrid waveform relaxation. X. Hu, C. Rodrigo, F.J. Gaspar. LINK

FILES

MATLAB Function to plot the two-grid convergence factors predicted by SAMA for a range of values of parameter \lambda from 2^(-12) to 2^(12) and for a fixed fractional order \delta given as input: FILE

LATEX file of paper 1: FILE