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Sensitivity of the solution of the Elder problem to density, velocity and numerical perturbations

机译:Elder问题解对密度,速度和数值扰动的敏感性

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In this paper the Elder problem is studied with the purpose of evaluating the inherent instabilities associated with the numerical solution of this problem. Our focus is first on the question of the existence of a unique numerical solution for this problem, and second on the grid density and fluid density requirements necessary for a unique numerical solution. In particular we have investigated the instability issues associated with the numerical solution of the Elder problem from the following perspectives: (ⅰ) physical instability issues associated with density differences; (ⅱ) sensitivity of the numerical solution to idealization irregularities; and, (ⅲ) the importance of a precise velocity field calculation and the association of this process with the grid density levels that is necessary to solve the Elder problem accurately. In the study discussed here we have used a finite element Galerkin model we have developed for solving density-dependent flow and transport problems, which will be identified as TechFlow. In our study, the numerical results of Frolkovic and de Schepper [Frolkovic, P. and H. de Schepper, 2001. Numerical modeling of convection dominated transport coupled with density-driven flow in porous media, Adv. Water Resour., 24, 63-72.] were replicated using the grid density employed in their work. We were also successful in duplicating the same result with a less dense grid but with more computational effort based on a global velocity estimation process we have adopted. Our results indicate that the global velocity estimation approach recommended by Yeh [Yeh, G.-T., 1981. On the computation of Darcian velocity and mass balance in finite element modelling of groundwater flow, Water Resour. Res., 17(5), 1529-1534.] allows the use of less dense grids while obtaining the same accuracy that can be achieved with denser grids. We have also observed that the regularity of the elements in the discretization of the solution domain does make a difference in obtaining a unique stationary solution for this problem. The results of our study also indicate that the density differences are critical in the solution of the Elder problem and that high density differences lead to the physical instability that is inherent with this problem. Other than the physical instability associated with the level of density differences used in the Elder problem, the following two points should be considered in solving the Elder problem in a consistent manner: (ⅰ) strict attention should be paid to the vertical grid Peclet number in developing the criteria for convergent grid selection; and, (ⅱ) with a globally continuous velocity calculation stable solutions can be obtained at lower grid densities.
机译:本文对Elder问题进行了研究,目的是评估与该问题的数值解相关的固有不稳定性。我们的重点首先是解决这个问题的唯一数值解的问题,其次是唯一数值解所必需的网格密度和流体密度要求。特别是,我们从以下角度研究了与Elder问题的数值解相关的不稳定性问题:(ⅰ)与密度差异有关的物理不稳定性问题; (ⅱ)数值解对理想化不规则性的敏感性; (ⅲ)精确速度场计算的重要性以及此过程与网格密度级别的关联,这对于准确解决Elder问题是必需的。在这里讨论的研究中,我们使用了有限元Galerkin模型,该模型是为解决与密度有关的流动和运输问题而开发的,该模型将被标识为TechFlow。在我们的研究中,Frolkovic和de Schepper的数值结果[Frolkovic,P. and H. de Schepper,2001.对流控制输运的数值模型以及密度驱动的多孔介质流动,Adv。使用他们在工作中使用的网格密度复制《水资源》,第24卷,第63-72页。我们还成功地通过密度较低的网格复制了相同的结果,但是基于我们采用的全局速度估计过程,需要进行更多的计算工作。我们的结果表明,Yeh [Yeh,G.-T.,1981年推荐了整体速度估算方法。关于地下水流量有限元建模中的Darcian速度和质量平衡的计算,水资源。 [Res。,17(5),1529-1534。]允许使用密度较小的网格,同时获得与密度较高的网格相同的精度。我们还观察到,求解域离散化中元素的规则性确实在为该问题获得唯一的平稳解中有所不同。我们的研究结果还表明,密度差异对于解决Elder问题至关重要,而高密度差异会导致该问题固有的物理不稳定性。除了与Elder问题中使用的密度差异级别相关的物理不稳定性外,在以一致的方式解决Elder问题时,还应考虑以下两点:(ⅰ)应严格注意垂直网格中的Peclet数制定收敛网格选择的标准;通过整体连续速度计算,可以在较低的网格密度下获得稳定的解。

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