A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves
A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves
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DOI：
10.1088/1361-6544/aa712d
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发表时间：
2017-05
期刊：
影响因子：
1.7
通讯作者：
N. Favrie;S. Gavrilyuk
中科院分区：
文献类型：
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作者：
N. Favrie;S. Gavrilyuk
提出了一种求解描述浅水色散波的Serre-Green-Naghdi（SGN）方程的新数值方法。从数学的角度来看，SGN方程是欧拉-拉格朗日方程的“主”拉格朗日提交的微分约束，这是质量守恒定律。求解SGN方程的一个主要数值挑战是在每个时刻解决椭圆问题。这是数值方法中最耗时的部分。这个想法是用一个单参数的“增广”拉格朗日函数族来代替“主”拉格朗日函数，这取决于更多的变量，相应的欧拉-拉格朗日方程是双曲的。在这种方法中，“主”拉格朗日量在一定限度内（例如，当相应的参数很大时）由增广拉格朗日量恢复。这样一个家庭的增广拉格朗日的选择提出和讨论。相应的双曲型方程组的数值求解由Goddham型方法。数值解与精确解的SGN方程进行比较。看来，在解决双曲型系统的计算时间是远远低于在椭圆算子被倒置的情况下。新的方法是适用于，特别是，“法弗尔波”的研究代表非静止的起伏孔后产生的流体流动的反射与自由表面在一个不动的壁。
A new numerical method for solving the Serre–Green–Naghdi (SGN) equations describing dispersive waves on shallow water is proposed. From the mathematical point of view, the SGN equations are the Euler–Lagrange equations for a ‘master’ lagrangian submitted to a differential constraint which is the mass conservation law. One major numerical challenge in solving the SGN equations is the resolution of an elliptic problem at each time instant. This is the most time-consuming part of the numerical method. The idea is to replace the ‘master’ lagrangian by a one-parameter family of ‘augmented’ lagrangians, depending on a greater number of variables, for which the corresponding Euler–Lagrange equations are hyperbolic. In such an approach, the ‘master’ lagrangian is recovered by the augmented lagrangian in some limit (for example, when the corresponding parameter is large). The choice of such a family of augmented lagrangians is proposed and discussed. The corresponding hyperbolic system is numerically solved by a Godunov type method. Numerical solutions are compared with exact solutions to the SGN equations. It appears that the computational time in solving the hyperbolic system is much lower than in the case where the elliptic operator is inverted. The new method is applied, in particular, to the study of ‘Favre waves’ representing non-stationary undular bores produced after reflection of the fluid flow with a free surface at an immobile wall.