Riesz空间分数阶扩散方程的分数阶中心差分加权离散格式

A Weighted Fractional Centered Difference Method for the Space Fractional Diffusion Equation Based on Riesz Derivatives

在有限区域内考虑带齐次Dirichlet边界条件的Riesz空间分数阶扩散方程的初边值问题,利用分数阶中心差分对空间方向进行离散,在时间方向上用隐式和显式Euler格式的加权平均进行离散,构造了空间2阶、时间γ阶(γ=1,2)的全离散加权差分格式.利用函数的单调性证明了当加权因子0≤θ≤1/2时差分离散格式是无条件稳定的,当1/2<θ≤1时差分离散格式是条件稳定的,并给出了稳定的条件.证明了相应差分离散格式的收敛性.用实际数值算例验证了差分离散格式的有效性.

In this paper,the space fractional diffusion equation with homogeneous Dirichlet boundary condition based on Riesz deriv- ative on a finite domain is considered. A weighted difference method with the numerical error O（h^2＋τ^γ） （γ=1 or 2） is derived, when the fractional centered difference is applied to obtain the finite difference approach in space and the weighted average method based on the implicit and explicit Euler method is proposed in the time direction. The stability analysis of this scheme is carried out by means of the monotonicity of function. It is proved that the method is unconditionally stable for 0≤θ≤1/2,and conditionally stable for 1/2〈θ≤1,here θ is the weighting parameter subjected to 0≤θ≤1. The convergence of the corresponding difference method is proved. Finally, a numerical example with known exact solution is examined to verify the effectiveness of the method.