拉格朗日坐标下(1+2)维浅水方程的对称性、守恒律与差分格式
Symmetries, conservation laws and difference schemes of the (1+2)-dimensional shallow water equations in Lagrangian coordinates
AI总结：
本文研究拉格朗日坐标下二维浅水方程的对称性与守恒律，构造了新的不变保守有限差分格式，其中部分格式具有质量和能量守恒律。
摘要：
本文研究了欧拉坐标和拉格朗日坐标下的二维浅水方程。给出了这些方程的拉格朗日形式和哈密顿形式。给出了将具有圆形或平面底部的二维浅水方程映射为多方指数为$γ=2$的多方气体动力学方程的变换。考虑了方程的群性质，并对椭圆抛物面底部地形情形进行了群分类。从离散化的角度讨论了拉格朗日坐标下二维浅水方程的性质。构造了方程及其一维约化的新的不变保守有限差分格式。这些格式的推导方式包括：扩展已知的一维格式，或基于能量守恒律形式的某些假设进行直接代数构造。在所提出的格式中，存在具有质量守恒律和能量守恒律的格式。
英文摘要：
The two-dimensional shallow water equations in Eulerian and Lagrangain coordinates are considered. Lagrangian and Hamiltonian formalism of the equations is given. The transformations mapping the two-dimensional shallow water equations with a circular or plane bottom into the gas dynamics equations of a polytropic gas with polytropic exponent $γ=2$ is represented. Group properties of the equations are considered, and the group classification for the case of the elliptic paraboloid bottom topography is performed. The properties of the two-dimensional shallow water equations in Lagrangian coordinates are discussed from the discretization point of view. New invariant conservative finite-difference schemes for the equations and their one-dimensional reductions are constructed. The schemes are derived either by extending the known one-dimensional schemes or by direct algebraic construction based on some assumptions on the form of the energy conservation law. Among the proposed schemes there are schemes possessing conservation laws of mass and energy.
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