Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the nonlocal property of fractional differential operators, numerical methods for space-fractional diffusion equations generate dense or even full coefficient matrices with complicated structures. Traditionally, these methods were solved via Gaussian elimination, which requires computational work of O ( N 3 ) per time step and O ( N 2 ) of memory to store where N is the number of spatial grid points in the discretization. The significant computational work and memory requirement of these methods makes a numerical simulation of three-dimensional space-fractional diffusion equations computationally prohibitively expensive. We present an alternating-direction implicit (ADI) finite difference formulation for space-fractional diffusion equations in three space dimensions and prove its unconditional stability and convergence rate provided that the fractional partial difference operators along x-, y-, z-directions commute. We base on the ADI formulation to develop a fast iterative ADI finite difference method, which has a computational work count of O ( N log N ) per iteration at each time step and a memory requirement of O ( N ) . We also develop a fast multistep ADI finite difference method, which has a computational work count of O ( N log 2 N ) per time step and a memory requirement of O ( N log N ) . Numerical experiments of a three-dimensional space-fractional diffusion equation show that these both fast methods retain the same accuracy as the regular three-dimensional implicit finite difference method, but have significantly improved computational cost and memory requirement. These numerical experiments show the utility of the fast method.