Parallel algorithms for nonlinear time-space fractional parabolic PDEs

In this paper, we develop a time stepping scheme for solving nonlinear time-space fractional partial differential equations (PDEs). In space, we use the matrix transfer technique to discretize the PDEs and obtain a system of nonlinear time-fractional differential equations. The developed scheme is similar to the Crank-Nicholson scheme for integer order PDEs and are shown to be of order 1+α in time where α is the order of the time derivative described in the Caputo sense. The solution of the PDE at any point i in the t-stencil depends not only on the solution at the point i−1 but on all previous solutions (memory effect). Thus, the implementation of schemes for such fractional PDEs, for long time interval, can be time consuming. This is basically due to the computation and re-computation of the history term at each time step. We lessen this computational time by implementing three parallel versions of the algorithm. The shared memory systems (OpenMP) and the distributed memory systems (MPI) are used for implementing the parallel algorithms. A third parallel version uses both the shared and distributed memory systems (Hybrid version). The advantages of the parallel algorithms over the sequential algorithm are discussed. The merits and demerits of each parallel versions of the algorithm over the others are examined. Numerical simulations are performed to support our theoretical observations.