Numerical simulation for the three-dimension fractional sub-diffusion equation

Fractional sub-diffusion equations have been widely used to model sub-diffusive systems. Most algorithms are designed for one-dimensional problems due to the memory effect in fractional derivative. In this paper, the numerical simulation of the 3D fractional sub-diffusion equation with a time fractional derivative of order $\alpha$ $(0<\alpha <1)$ is considered. A fractional alternating direction implicit scheme (FADIS) is proposed. We prove that FADIS is uniquely solvable, unconditionally stable and convergent in $H^1$ norm by the energy method. A numerical example is given to demonstrate the efficiency of FADIS.