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Yin, Xiucao; Fang, Shaomei; Guo, Changhong

Alternating-direction implicit finite difference methods for a new two-dimensional two-sided space-fractional diffusion equation. (English) Zbl 1448.65125

Adv. Difference Equ. 2018, Paper No. 389, 17 p. (2018).
Summary: 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.

Cited in 3 Documents

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
26A33 Fractional derivatives and integrals
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics

Keywords:

two-dimensional two-sided space-fractional diffusion equations; shifted left Grünwald formula; standard right Grünwald formula; ADI methods; Richardson extrapolation
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