基于R-L定义的分数微分对流-弥散方程有限元解

A Riemann-Liouville definition based finite element solution for fractional advection-dispersion equation

分数微分对流-弥散方程(Fractional Advection-Dispersion Equation,FADE)是一种用于模拟多孔介质中溶质非费克迁移的新模型,然而由于分数微分定义的复杂性,仅能够获得特定的定解条件下FADE模型解析解.推导出了基于Riemman-Liouville(R-L)定义的FADE模型有限元解,当分数阶微分算子α=2时,该解与传统对流-弥散方程的有限元解相同.与Meerschart和Tadjeran(2004)的有限差分解及FADE模型的解析解的模拟结果相比,本文的有限元解在很大程度上能降低数值弥散现象,但当空间离散节点数目较大时(N>100),都会产生质量不守恒的现象.通过模拟结果和相关文献的分析比较得出,FADE模型的这种质量不守恒问题是由于R-L定义本身所引起的,解决该问题需要对FADE模型的数值解做进一步的研究.

The fractional advection-dispersion equation （FADE） with non-local properties is a promising approach to describe the non-Fickian transport of contaminants in porous media. However, the analytical solutions of FADE can only be achieved with specific initial and boundary conditions. We presented a finite element （FEM） solution for the one-dimensional FADE based on the Riemann-Liouville （R-L） definition of the fractional derivative, which are convergences to the finite element solution for traditional ADE when α equals 2. The FEM solution is used to comparing the finite difference （FDM） solution and the analytical solution of the FADE. Results showed that the FEM scheme is better than the finite element solution with less numerical dispersion. Considerable mass conservancy problem occurs either in the FEM or FDM when the number of spatial discretization is larger than 100. This problem is caused by the R-L definition of fractional derivative. Further research is still needed to solve the mass conservancy problem occurring in the numerical solution of FADE.