Grid convergence for numerical solutions of stochastic moment equations of groundwater flow
Grid convergence for numerical solutions of stochastic moment equations of groundwater flow
复制标题
地下水流随机矩方程数值解的网格收敛
DOI：
10.1007/s00477-019-01719-6
复制
发表时间：
2019
影响因子：
4.2
通讯作者：
Ackerer Philippe
中科院分区：
文献类型：
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作者：
Xia Chuan An;Guadagnini Alberto;Hu Bill X;Riva Monica;Ackerer Philippe
我们根据(a)收敛速度/顺序和(b)网格收敛指数（GCI）对网格收敛研究的结果进行定性和定量评估，GCI与稳态地下水流动的矩方程（MEs）的数值解相关。后者近似为二阶（根据水力传导性的自然对数Y的标准偏差）。我们考虑(1)Riva等人（Transp Porous Med 45(1):139 - 193,2001）在随机非均质导电性场中稳态径向流动的解析解，作为参考；(2)水头和流量的（集合）均值和（co）方差所满足的MEs的数值解。基于45个离散程度不同的数值网格，我们发现水头的均值和方差以及流量横向分量的方差具有超线性收敛速率，径向流量的方差具有亚线性收敛速率。我们对GCI的估计表明，要准确计算头和通量的均值和（co）方差，需要对每个y相关长度至少包含8个网格元素的空间离散化，而要准确表示平均头的二阶分量，则需要更精细的离散化。
We provide qualitative and quantitative assessment of the results of a grid convergence study in terms of (a) the rate/order of convergence and (b) the grid convergence index, GCI, associated with the numerical solutions of moment equations (MEs) of steady-state groundwater flow. The latter are approximated at second order (in terms of the standard deviation of the natural logarithm,Y, of hydraulic conductivity). We consider (1) the analytical solutions of Riva et al. (Transp Porous Med 45(1):139–193, 2001) for steady-state radial flow in a randomly heterogeneous conductivity field, which we take as references; and (2) the numerical solutions of the MEs satisfied by the (ensemble) mean and (co)variance of hydraulic head and fluxes. Based on 45 numerical grids associated with differing degrees of discretization, we find a supra-linear rate of convergence for the mean and (co)variance of hydraulic head and for the variance of the transverse component of fluxes, the variance of radial fluxes being characterized by a sub-linear convergence rate. Our estimated values of GCI suggest that an accurate computation of mean and (co)variance of head and fluxes requires a space discretization comprising at least 8 grid elements per correlation length ofY, an even finer discretization being required for an accurate representation of the second-order component of mean heads.