对流扩散方程的三阶迎风格式的数值摄动高精度重构

利用高智提出的数值摄动算法,把求解对流扩散方程常用三阶迎风格式(3-UDS)(粘性项和对流项分别用二阶中心格式和3-UDS离散)进行了高精度重构,包括使用离散单元内所有节点的全域重构和分别使用上下游节点的上下游重构,得到两类新的更高阶精度迎风差分格式,称为高的迎风差分格式(记作GUDS)。讨论了GUDS的数学性质,GUDS比原来的3-UDS精度显著提高;全域重构的GUDS和3-UDS均为条件稳定,一些上下游重构GUDS为绝对稳定。本文通过稳定性分析和四个算例(一维常系数、变系数、非线性及二维变系数对流扩散方程)的计算证实了GUDS的优良性质。上下游重构GUDS为避免在3-UDS中使用人工粘性提供了一条有效途径,适合于求解高Reynolds数线性和非线性问题。

A finite-element algorithm for modeling variably saturated flows

A general numerical algorithm in the context of finite element scheme is developed to solve Richards’ equation, in which a mass-conservative, modified head based scheme (MHB) is proposed to approximate the governing equation, and mass-lumping techniques are used to keep the numerical simulation stable. The MHB scheme is compared with the modified Picard iteration scheme (MPI) in a ponding infiltration example. Although the MHB scheme is a little inferior to the MPI scheme in respect of mass balance, it is superior in convergence character and simplicity. Fully implicit, explicit and geometric average conductivity methods are performed and compared, the first one is superior in simulation accuracy and can use large time-step size, but the others are superior in iteration efficiency. The algorithm works well over a wide variety of problems, such as infiltration fronts, steady-state and transient water tables, and transient seepage faces, as demonstrated by its performance against published experimental data. The algorithm is presented in sufficient detail to facilitate its implementation.