分数阶对流——弥散方程的数值求解

对严格的时间分数阶对流--弥散方程和严格的空间分数阶对流--弥散方程分别建立了差分格式,并用所建立的两个差分格式对同一理想算例进行了求解.通过对分数阶导数取不同的参数值,得到一系列结果,分析了不同分数阶导数描述的反常扩散现象及其变化规律,并和传统的整数阶对流--弥散方程的求解结果进行了对比.当时间分数阶对流--弥散方程和空间分数阶对流--弥散方程的分数阶导数的参数分别取整数值时,时间分数阶对流--弥散方程、空间分数阶对流--弥散方程和传统整数阶对流--弥散方程的计算结果相同,表明本文提出的对时间分数阶对流--弥散方程和空间对流--弥散方程数值求解方法是可行的,且整数阶对流--弥散方程是分数阶对流--弥散方程的特殊情况.和正常扩散相比,时间分数阶对流--弥散方程中分数阶导数的参数值越小,溶质扩散得越慢,表现为拖尾分布:空间分数阶对流--弥散方程中分数阶导数的参数值越小,溶质扩散得越快,表明空间的非局域性相关性越强.

Numerical Solutions of Fractional Advection-Dispersion Equations

Two numerical schemes were developed for both spatially and tenporally fractional advection-dispersion equations , respectively. By applying both schemes to the same test case, we analyzed the phenomena of abnormal diffusion resulting from variations of the fractional order of relevant derivatives, and confirmed the accurate match between the solutions of our new schemes and the traditional one. If the fractional orders of spatially and tenporally fractional advection-dispersion equations are integer, the calculated results of both spatially and tenporally fractional advection-dispersion equations are the same as the calculated result of integer order advection-dispersion equation, which indicates that the two numerical schemes we developed are feasible, and the inter order advection-dispersion is a special case of fractional advection-dispersion equation. Compared with normal diffusion, the lower the fractional order of time fractional advection-dispersion equation is, the more slowly the solute diffuse, which represents long tail. Correspondingly, the lower the fractional order of space fractional advection-dispersion equation is, the more rapidly the solute diffuse, which demonstrates that the non-local correlation on space is more intense.