非均匀海底浅水方程的Hamilton正则化
Hamiltonian regularisation of shallow water equations with uneven bottom
摘要：
非线性双曲型守恒律方程的正则化对于获得弱解的唯一性和精确的数值模拟都具有重要意义。在最近的工作中，前两个作者提出了一个所谓的哈密顿正则化的非线性浅水和等熵欧拉方程。这种方法的特点是，正则化的解决方案是实现不添加任何人工耗散或ispersion。正则化系统具有哈密顿结构，因此，形式上保持相应的能量泛函。在本文中，我们概括了这种方法，浅水波一般，可能随时间变化，底部。用连续Galerkin方法数值求解了该系统，并将其解与经典浅水和色散Serre-Green-Naghdi方程的类似解进行了比较。数值计算结果证实，在水深变化的存在下，色散和耗散的影响。
英文摘要：
The regularisation of nonlinear hyperbolic conservation laws has been a problem of great importance for achieving uniqueness of weak solutions and also for accurate numerical simulations. In a recent work, the first two authors proposed a so-called Hamiltonian regularisation for nonlinear shallow water and isentropic Euler equations. The characteristic property of this method is that the regularisation of solutions is achieved without adding any artificial dissipation or ispersion. The regularised system possesses a Hamiltonian structure and, thus, formally preserves the corresponding energy functional. In the present article we generalise this approach to shallow water waves over general, possibly time-dependent, bottoms. The proposed system is solved numerically with continuous Galerkin method and its solutions are compared with the analogous solutions of the classical shallow water and dispersive Serre-Green-Naghdi equations. The numerical results confirm the absence of dispersive and dissipative effects in presence of bathymetry variations.
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