Geodesic
Efficient Parareal algorithm for solving time-fractional diffusion equation
Dalʹnevostočnyj matematičeskij žurnal, Tome 22 (2022) no. 2, pp. 245-251.

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The work is devoted to developing efficient parallel algorithms for solving the initial boundary problem for the time-fractional diffusion equation. Traditional approaches to parallelization are based on the space domain decomposition. In contrast, the parareal method is based on the time domain decomposition and an iterative predictor-corrector procedure. The fast solver on a coarse grid is used to construct the initial approximations for subtasks (solved by accurate solvers on finer grids) and for correcting the solutions of subtasks. The subtasks may be solved independently for each subinterval of time. This allows one to implement the efficient parallel algorithms for various high-performance architectures. Currently, this method is widely used for problems for classical differential equations with integer orders. But it is much less commonly used for the fractional equations. In this work, the parareal algorithm for solving the initial boundary problem for the time-fractional diffusion equation is implemented using the OpenMP technology for multicore processors. The numerical experiments are performed to estimate the efficiency of parallel implementation and compare the parareal algorithm with the traditional space domain decomposition. 
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M. A. Sultanov; V. E. Misilov; Y. Nurlanuly. Efficient Parareal algorithm for solving time-fractional diffusion equation. Dalʹnevostočnyj matematičeskij žurnal, Tome 22 (2022) no. 2, pp. 245-251. http://geodesic.mathdoc.fr/item/DVMG_2022_22_2_a19/

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