Algorithm Analysis, Code Analysis, Code Vectorization, and Code Parallelization of a Fast Finite Difference Method for Space Fractional Diffusion Equations in Three Space Dimensions

Fractional diffusion equations can be used to model phenomena with anomalous diffusion which cannot be accurately modeled by standard second-order diffusion equations. Due to the nonlocal property of fractional differential operators, numerical methods for space-fractional diffusion equations generate full coefficient matrices requiring storage on the order of O( N2), where N is the number of spatial grid points in the discretization. These methods are traditionally solved with Gaussian elimination requiring a computational cost on the order of O(N3) per time step. These methods have a large computational workload as well as a large memory requirement, making for slow run times. A fast multistep alternating-direction implicit (ADI) finite difference method which has a computational work count of O(Nlog2N) per time step and a memory requirement of O(NlogN) is presented. Previous numerical experiments of a three-dimensional space-fractional diffusion equation show that this method retains the same accuracy as the regular three-dimensional implicit finite difference method, but with significantly improved computational cost and memory requirement. This fast multistep ADI method will be updated and optimized and compared to previous run times.