Riesz回火分数阶平流-扩散方程的隐式中点方法

THE IMPLICIT MIDPOINT METHOD FOR RIESZ TEMPERED FRACTIONAL ADVECTION-DIFFUSION EQUATION

本文针对Riesz回火分数阶平流-扩散方程,采用隐式中点方法离散一阶时间偏导数,用修正的二阶Lubich回火差分算子逼近Riesz空间回火分数阶偏导数,并对平流项采用中心差商进行离散,构造出新的数值方法,获得了数值方法的稳定性和收敛性,该方法的收敛阶在空间和时间方向均达到二阶精度.数值试验验证了数值方法的有效性.

In this paper,an implicit midpoint method is applied to discretize the first order time partial derivative,the modified second-order Lubich tempered difference operator is used to approximate Riesz space tempered fractional partial derivative,and the central difference formula is utilizing to discrete the advection term,a numerical scheme is constructed for solving Riesz tempered fractional advection-diffusion equation.The stability and convergence of the numerical scheme is established,and the convergence order of numerical scheme can reach two order accuracy on temporal and spatial directions respectively.The numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the schemes.