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

Xiucao Yin,Changhong Guo,Shaomei Fang

doi:10.1186/s13662-018-1836-z

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.

Highlights

IntroductionAccording to the principle of conservation of mass, the equation of continuity form is given by
According to the principle of conservation of mass, the equation of continuity form is given by∂u(x, t) ∂Q(x, t) + = f (x, t), (1.1) ∂t∂x where u(x, t) is the distribution function of the diffusing quantity, Q(x, t) is the diffusion flux, and f (x, t) is the source term
Theorem 1 The implicit Euler method defined by Eq (3.2) is consistent with model Eq (1.5) of the order O(τ + h1 + h2)

Summary

Introduction

According to the principle of conservation of mass, the equation of continuity form is given by. [13] discussed the practical alternating-directions implicit method to solve the twodimensional two-sided space fractional convection diffusion equation on a finite domain. Proposed a Crank–Nicolson alternating-direction implicit Galerkin–Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equations. In the numerical aspect of these two-sided space-fractional diffusion equations in one dimension, Chen et al [1] developed a fast semi-implicit difference method for a nonlinear one-dimensional two-sided space-fractional diffusion equation with variable diffusivity coefficients. The study on the finite difference method computation of these two-sided space-fractional diffusion equations in two dimensions is limited This motivates us to develop the alternating-direction finite difference methods for this two-dimensional twosided space-fractional diffusion equation in this paper.

Notations and properties

ADI–CN method and its theory analysis

Conclusions