An improved method for inhomogeneous space grid in the simulation of unsaturated flow
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摘要:
Richards方程在非饱和渗流模拟及其他相关领域应用广泛。在数值求解过程中，可以采用有限差分方法进行数值离散并迭代求解，为了获得较可靠的数值解，常规的均匀网格空间步长往往是较小的。在一些不利数值条件下，如入渗于干燥土壤，迭代计算费时甚至精度也不能得到很好改善。因此，文章提出Chebyshev空间网格改进方法，结合有限差分方法对Richards方程进行数值离散以获得线性方程组，并通过经典的Picard迭代方法进行迭代求解线性方程组以得到Richards方程的数值解。通过均质土和分层土2个不利情况下的非饱和渗流算例，又结合模型解析解和软件Hydrus-1D，对比研究了改进网格方法与均匀网格方法获得数值解的精度。结果表明，提出的Chebyshev网格方法相较于传统的均匀网格，可以在较少的节点数下获得较高的数值精度，又具有较小的计算开销，有较好的应用前景。
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关键词:
- Richards方程 /
- 有限差分 /
- 均匀网格 /
- Chebyshev网格 /
- 数值精度
Abstract:The Richards’ equation is widely used in the simulation of unsaturated flow and related fields. In the numerical solution process, the finite difference method can be used to carry out numerical discretization and iterative calculation. However, in order to obtain a more reliable numerical solution, the space step size of a conventional uniform grid is often small. For some unfavorable numerical conditions, such as infiltration into dry soil, iterative calculation is time-consuming and even the accuracy cannot be improved very well. Therefore, an improved method is proposed by using the Chebyshev space grid, which combines the finite difference method to numerically discretize the Richards’ equation to obtain linear equations. Then the classic Picard iterative method is used to iteratively solve the linear equations to obtain the numerical solutions of the Richards’ equations. Through two examples of unsaturated flow under unfavorable conditions for homogeneous soil and layered soil, combined with the analytical solution of the model and the software Hydrus-1D, the accuracy of the numerical solution obtained by the improved grid method and the uniform grid method is compared and examined. The results show that the proposed Chebyshev grid method can obtain higher numerical accuracy with a smaller number of nodes than the traditional uniform grid, and the computational cost is smaller. This method has a good application prospect.
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Key words:
- Richards’ equation /
- finite difference /
- uniform grid /
- Chebyshev grid /
- numerical accuracy
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表 1 t=5 h时的数值精度
Table 1. Numerical accuracy at t=5 h
条件 RSE RE/% N 均匀网格法 Chebyshev网格法 均匀网格法 Chebyshev网格法 100 0.11 4.7×10−3 5.01 0.0502 150 0.10 2.6×10−3 4.53 0.1217 200 0.09 2.1×10−3 4.05 0.1476 
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表 2 饱和土壤的水力传导系数
Table 2. Hydraulic conductivity values for saturated soils
土壤类型 饱和水力传导系数/（m·s−1） 无杂质的砂石 10−2～1 粗砂 10−4～10−2 细砂 10−5～10−4 粉土 10−7～10−5 黏土 <10−8
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表 3 工况1至8的水力传导系数
Table 3. Hydraulic conductivity for cases 1 to 8
工况数 1 2 3 4 5 6 7 8 Ks1/
（m·s−1）10−1 10−1 10−1 10−1 10−1 10−1 10−1 10−1 Ks2/
（m·s−1）10−2 10−3 10−4 10−5 10−6 10−7 10−8 10−9
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表 4 数值精度比较
Table 4. Comparison of the numerical accuracy
方法 RSE RE/% MRE/% 均匀网格 
2.81 45.1 Chebyshev网格 
2.20 34.2
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