Alternating-direction implicit finite difference methods for a new two-dimensional two-sided space-fractional diffusion equation

Abstract

According to the principle of conservation of mass and the fractional Fick's law, a new two-sided space-fractional diffusion equation was obtained. In this paper, we present two accurate and efficient numerical methods to solve this equation. First we discuss the alternating-direction finite difference method with an implicit Euler method (ADI-implicit Euler method) to obtain an unconditionally stable first-order accurate finite difference method. Second, the other numerical method combines the ADI with a Crank-Nicolson method (ADI-CN method) and a Richardson extrapolation to obtain an unconditionally stable second-order accurate finite difference method. Finally, numerical solutions of two examples demonstrate the effectiveness of the theoretical analysis.

Finite Difference Method for Two-Sided Two Dimensional Space Fractional Convection-Diffusion Problem with Source Term

“…There are numerical schemes that used to solve two-dimension space fractional diffusion problems such as the alternating direction implicit (ADI) method [25][26][27][28][29][30], the Galerkin finite element method [31], the finite volume method [32] and the kronecker product splitting method [33]. ADI and CN-ADI spectral methods are used to solve two-dimensional Riesz space fractional diffusion equation with a non-linear reaction term with respect to their error estimates have been discussed (see References [34,35]).…”

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

“…In multidimensional case, lowering of computational complexity is often achieved using splitting schemes (Chen and Li 2016;Yin et al 2018;Samarskii 2001;Bogaenko et al 2017) that reduce multidimensional problems to the series of one-dimensional ones. In addition to the direct effect of complexity reduction, significant reduction in the connectivity between computation blocks makes it possible to apply parallelization techniques efficiently in this case.…”