二维输运方程高精度数值模拟

High-accuracy numerical simulation of 2D transport problems

采用剖开算子法,把二维输运问题剖分为两个子初值问题(对流分步、扩散分步)。在任意三角形网格中,分别对不同性质的算子采用各自适合的算法,即采用特征线法求解对流分步,采用半隐式有限元法求解扩散分步。重点探讨了对流插值问题,给出了一种完全对称三次插值模式,有效地减少了数值阻尼。为了克服高阶插值数值震荡问题,计算中保证了函数及其一阶偏导数连续。算例表明,数值方法模拟结果与精确解吻合较好。该算法在求解输运方程(包括纯对流输运方程)时,既能有效减少数值阻尼,也能保证计算中不出现数值震荡。

An operator-splitting algorithm for the two-dimensional convection-dispersion equation is developed. The governing equations are split into two successive initial value problems, which include a pure convection problem and a pure dispersion problem. For any arbitrary triangle calculation grids, suitable algorithms will be applied to resolve the problems at each grid point according operator characteristics. In the case of the convection problem, the method of characteristics can be applied. While for the dispersion problem, a semi-implicit finite element method will be em- ployed. The issue concerning interpolation in convection problem will be discussed in detail, and a cubic interpolation method will be proposed to reduce the numerical damping effect. To overcome the numerical oscillation problem in high-order interpolation, it is required to keep the continuity of function and its partial derivatives. The testing results show that the numerical simulations of this study agree well with analytical solutions. The proposed operator-splitting algorithm can significantly reduce the numerical damping effect and overcome the numerical oscillation problem in high-order interpolation when solving pure convective transport problems.