一维Green-Naghdi方程的对称性、守恒律、不变解与差分格式
Symmetries, conservation laws, invariant solutions and difference schemes of the one-dimensional Green-Naghdi equations
AI总结：
本文在Lagrangian坐标下对一维Green-Naghdi方程进行Lie群分类，利用Noether定理导出不平坦底部地形下的守恒律，并构造保持对称性和守恒律的差分格式，通过精确行波解进行数值验证。
摘要：
本文致力于研究描述流体在不平坦底部地形上流动行为的一维Green-Naghdi方程的Lie群性质。底部地形以两种方式纳入Green-Naghdi方程：经典Green-Naghdi形式和同阶近似形式。研究在Lagrangian坐标下进行，这使得能够找到所分析方程的Lagrangian函数。给出了两种Green-Naghdi方程情形关于底部地形的完整群分类。应用Noether定理，结合所得的Lagrangian函数和群分类，得到了具有不平坦底部地形的一维Green-Naghdi方程的守恒律。构造了保持原方程对称性和守恒律的差分格式。对所发展的格式进行了分析。这些格式通过精确行波解的例子进行了数值检验。
英文摘要：
The paper is devoted to the Lie group properties of the one-dimensional Green-Naghdi equations describing the behavior of fluid flow over uneven bottom topography. The bottom topography is incorporated into the Green-Naghdi equations in two ways: in the classical Green-Naghdi form and in the approximated form of the same order. The study is performed in Lagrangian coordinates which allows one to find Lagrangians for the analyzed equations. Complete group classification of both cases of the Green-Naghdi equations with respect to the bottom topography is presented. Applying Noether's theorem, the obtained Lagrangians and the group classification, conservation laws of the one-dimensional Green-Naghdi equations with uneven bottom topography are obtained. Difference schemes which preserve the symmetries of the original equations and the conservation laws are constructed. Analysis of the developed schemes is given. The schemes are tested numerically on the example of an exact traveling-wave solution.
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