We present a novel hyperbolic reformulation of the Serre-Green-Naghdi (SGN) model for the description of dispersive water waves. Contrarily to the classical Boussinesq-type models, it contains only first order derivatives, thus allowing to overcome the numerical difficulties and the severe time step restrictions arising from higher order terms. The proposed model reduces to the original SGN model when an artificial sound speed tends to infinity. Moreover, it is endowed with an energy conservation law from which the energy conservation law associated with the original SGN model is retrieved when the artificial sound speed goes to infinity. The governing partial differential equations are then solved at the aid of high order ADER discontinuous Galerkin finite element schemes. The new model has been successfully validated against numerical and experimental results, for both flat and non-flat bottom. For bottom topographies with large variations, the new model proposed in this paper provides more accurate results with respect to the hyperbolic reformulation of the SGN model with the mild bottom approximation recently proposed in "C. Escalante, M. Dumbser and M.J. Castro. An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes, Journal of Computational Physics 2018".

中文翻译：

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用于一般底部地形的 Serre-Green-Naghdi 模型的双曲线重构

我们提出了一种新的 Serre-Green-Naghdi (SGN) 模型的双曲线重构，用于描述分散的水波。与经典的 Boussinesq 型模型相反，它只包含一阶导数，因此可以克服高阶项引起的数值困难和严格的时间步长限制。当人工声速趋于无穷大时，所提出的模型简化为原始 SGN 模型。此外，它被赋予了能量守恒定律，当人工声速达到无穷大时，可以从中检索出与原始 SGN 模型相关的能量守恒定律。然后在高阶 ADER 不连续伽辽金有限元方案的帮助下求解控制偏微分方程。新模型已成功通过数值和实验结果验证，适用于平底和非平底。对于变化较大的底部地形，本文提出的新模型在“C. Escalante、M. Dumbser 和 MJ Castro.An Effective用于分散非静水波的双曲弛豫系统及其高阶不连续伽辽金方案的解决方案，计算物理学杂志 2018"。