时间分数阶对流-扩散方程的有限差分法

The finite difference scheme for the time fractional advection-diffusion equation

时间分数阶对流-扩散方程可以用来模拟由传统的对流-扩散方程演变而来的反常扩散方程.本文针对一类时间分数阶对流-扩散方程提出了一个新的隐式差分格式,时间分数阶导数采用直接离散,空间导数采用中心差分格式离散,讨论了差分解的存在唯一性,并利用能量范数证明了该格式的无条件稳定性、收敛性,分析了收敛阶.数值试验验证了该格式的有效性.

Time fractional advection-diffusion equation,which is different from traditional convection-diffusion equation,can be utilized to simulate time-related anomalous diffusion.In this paper,We propose a new implicit difference approximation to solve a time fractional advection-diffusion equation.The time fractional derivative is directly discretized,the spatial derivative is discretized by central difference scheme.We discuss the existence and uniqueness of the solution of the scheme.Using the energy norm,we prove the unconditional stability,convergence of the scheme,and analyze its convergence order.Finally,a numerical experiment shows that the scheme is efficient.