基于双曲分解和CABARET格式的多层浅水方程新数值算法

IF 0.7 Q4 OCEANOGRAPHY
V. M. Goloviznin, Pavel A. Maiorov, Petr A. Maiorov, A. V. Solovjov
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引用次数: 5

摘要

意图本文致力于描述具有自由表面和变密度的不可压缩流体动力学问题的流体静力学近似的一种新的数值求解方法。方法和结果。该算法基于双曲分解方法,即将多层模型表示为通过层界面的反作用力相互作用的单层模型的总和。作用在各层上下界面上的力被解释为不破坏各层方程组双曲性的外力。显式CABARET格式用于求解每层密度可变的双曲方程组。该方案具有二阶近似和时间可逆性。其特征在于增加了自由度:除了参考计算单元中心的保守型变量外,还应用了与这些单元垂直边缘中间相关的通量型变量。多层浅水方程组不是无条件双曲的,当双曲性丢失时,它就变为不适定的。双曲分解并不能消除原多层浅水方程组的不正确性。为了正则化数值解,提出了以下一组工具:过滤每个时间步长的流量变量;压力梯度的超隐式近似;线性人工粘度和向欧拉-拉格朗日(SEL)变量的转换,从而导致层之间的质量和动量交换。这种向SEL变量的转换是在很大程度上稳定数值解的基本工具。其余的技巧是辅助技巧,用于微调。结论。结果表明,正则化和保证问题的稳定性不仅需要在每个时间步长重建计算网格,还需要应用流型变量滤波和模拟湍流混合的人工粘度。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
New Numerical Algorithm for the Multi-Layer Shallow Water Equations Based on the Hyperbolic Decomposition and the CABARET Scheme
Purpose. The present article is devoted to describing a new method of numerical solution for hydrostatic approximation of incompressible hydrodynamic problems with free surfaces and variable density. Methods and Results. The algorithm is based on the hyperbolic decomposition method, i. e. representation of a multilayer model as a sum of the one-layer models interacting by means of the reaction forces through the layers’ interfaces. The forces acting on the upper and lower interfaces of each layer are interpreted as the external ones which do not break hyperbolicity of the equations system for each layer. The explicit CABARET scheme is used to solve a system of hyperbolic equations with variable density in each layer. The scheme is of the second approximation order and the time reversibility. Its feature consists in the increased number of freedom degrees: along with the conservative-type variables referred to the centers of the calculated cells, applied are the flux-type variables related to the middle of the vertical edges of these cells. The system of the multilayer shallow water equations is not unconditionally hyperbolic, and in case hyperbolicity is lost, it becomes ill-posed. Hyperbolic decomposition does not remove incorrectness of the original system of the multilayer shallow water equations. To regularize the numerical solution, the following set of tools is propose: filtration of the flow variables at each time step; super-implicit approximation of the pressure gradient; linear artificial viscosity and transition to the Euler-Lagrangian (SEL) variables that leads to the mass and momentum exchange between the layers. Such transition to the SEL variables is the basic tool for stabilizing numerical solution at large times. The rest of the tricks are the auxiliary ones and used for fine tuning. Conclusions. It is shown that regularizing and guaranteeing the problems’ stability requires not only reconstruction of the computational grid at each time step, but also application of the flow-type variables’ filtering and the artificial viscosity simulating turbulent mixing.
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来源期刊
Physical Oceanography
Physical Oceanography OCEANOGRAPHY-
CiteScore
1.80
自引率
25.00%
发文量
8
审稿时长
24 weeks
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