Parallel Implementation of the Algorithm for Solving a Partial Differential Equation with a Fractional Derivative in the Sense of Riemann-Liouville

The paper is devoted to the numerical solution of a two-dimensional partial differential equation with a fractional derivative in the sense of Riemann-Liouville. This equation is of great applied importance and underlies many multiphysical simulators for modeling the distribution of pollutants in the atmosphere, fluid motion in a fractured porous medium with fractal fracture geometry, and many others. A two-step implicit computational algorithm for solving the problem is proposed, which leads to the need to solve a large number of systems of linear algebraic equations with dense matrices. To solve the latter, three iterative Krylov subspace algorithms are considered, namely, generalized method of minimal residuals with restarts, quasiminimal residual method, and induced dimension reduction method. A method for organizing parallel computations using CPU threads is proposed. The proposed method is tested on a two-dimensional model problem for an equation with fractional derivatives. An equation of this type typically arises when considering non-Darcian fluid flow in a fractured porous medium with a fractal geometry of fractures. The results of calculations for different values of the fractional derivative exponent and mesh configurations are presented. It is shown that this approach is able to increase the performance of the algorithm by more than 3.6 times.