Parallel finite-difference algorithms for three-dimensional space-fractional diffusion equation with $\psi$-Caputo derivatives

Abstract

The paper deals with the issues of parallel computations’ organization while solving three-dimensional space-fractional diffusion equation with the $\psi$-Caputo derivatives using finite difference schemes. For an implicit scheme and locally one-dimensional splitting scheme, we present parallel algorithms for distributed memory systems that use one-dimensional block and red–black data partitioning. To reduce the order of algorithms’ computational complexity, we use an approach based on the expansion of integral operator’s kernel into series. We present the theoretical estimates of parallel algorithms’ performance and the results of computational experiments conducted on a testing problem that has an analytical solution for the case of the Caputo–Katugampola derivative. The results of the experiments show close-to-linear parallelization efficiency of one-dimensional splitting scheme with block partitioning and inefficiency of red–black partitioning in this case. For the implicit scheme, the scalability of parallel algorithms is weak and the use of red–black partitioning is more efficient than the use of block partitioning when running on a small number of computational resources.