On the Quality of Velocity Interpolation Schemes for Marker-in-Cell Method and Staggered Grids
On the Quality of Velocity Interpolation Schemes for Marker-in-Cell Method and Staggered Grids
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DOI：
10.1007/s00024-016-1431-8
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发表时间：
2015-12
影响因子：
2
通讯作者：
A. E. Pusok;B. Kaus;A. Popov
中科院分区：
文献类型：
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作者：
A. E. Pusok;B. Kaus;A. Popov
单元中标记方法通常被认为是对异质非扩散性质（即，岩石类型或组成）的地球动力学问题。在这种方法中，拉格朗日点携带的成分信息平流与周围的速度场的欧拉网格。然而，从网格点到标记位置的速度内插通常在不考虑内插位置处的速度场的发散的情况下执行（即，非保守）。当存在强速度梯度时，这种内插方案可以引起标记的非物理聚类（Journal of Computational Physics 166：218-252，2001），并且这可能最终导致空网格单元，这是对单元中标记方法的严重数值违反。为了以低计算成本弥补这一点，Jenny et al.（Journal of Computational Physics 166：218-252，2001）和Meyer和Jenny（Proceedings in Applied Mathematics and Mechanics 4：466-467，2004）提出了一种用于2-D交错网格的简单、保守的速度插值方案，而Wang等人提出了一种用于2-D交错网格的速度插值方案。（Geochemistry，Geophysics，Geosystems 16（6）：2015-2023，2015）将公式化扩展到3-D有限元方法。在这里，我们适应这个配方为3-D交错网格（校正插值），我们报告的质量为2-D和3-D交错网格的各种速度插值方法。我们测试的插值方案结合不同的平流计划不可压缩的Stokes问题的强速度梯度，使用有限差分法离散。我们的研究结果表明，一个保守的制定减少了分散和集群的标记，最大限度地减少非物理标记控制的地球动力学模型的需要。
The marker-in-cell method is generally considered a flexible and robust method to model the advection of heterogenous non-diffusive properties (i.e., rock type or composition) in geodynamic problems. In this method, Lagrangian points carrying compositional information are advected with the ambient velocity field on an Eulerian grid. However, velocity interpolation from grid points to marker locations is often performed without considering the divergence of the velocity field at the interpolated locations (i.e., non-conservative). Such interpolation schemes can induce non-physical clustering of markers when strong velocity gradients are present (Journal of Computational Physics 166:218–252, 2001) and this may, eventually, result in empty grid cells, a serious numerical violation of the marker-in-cell method. To remedy this at low computational costs, Jenny et al. (Journal of Computational Physics 166:218–252, 2001) and Meyer and Jenny (Proceedings in Applied Mathematics and Mechanics 4:466–467, 2004) proposed a simple, conservative velocity interpolation scheme for 2-D staggered grid, while Wang et al. (Geochemistry, Geophysics, Geosystems 16(6):2015–2023, 2015) extended the formulation to 3-D finite element methods. Here, we adapt this formulation for 3-D staggered grids (correction interpolation) and we report on the quality of various velocity interpolation methods for 2-D and 3-D staggered grids. We test the interpolation schemes in combination with different advection schemes on incompressible Stokes problems with strong velocity gradients, which are discretized using a finite difference method. Our results suggest that a conservative formulation reduces the dispersion and clustering of markers, minimizing the need of unphysical marker control in geodynamic models.