In this paper, we propose a unified and high order accurate fully-discrete one-step ADER Discontinuous Galerkin method for the simulation of linear seismic waves in the sea bottom that are generated by the propagation of free surface water waves. A hyperbolic reformulation of the Serre-Green-Naghdi model for nonlinear dispersive free surface flows is coupled with a first order velocity-stress formulation for linear elastic wave propagation in the sea bottom. Cartesian non-conforming meshes are defined and the coupling is achieved by an appropriate time-dependent pressure boundary condition in the three-dimensional domain for the elastic wave propagation, where the pressure is a combination of hydrostatic and non-hydrostatic pressure in the water column above the sea bottom. The use of a first order hyperbolic reformulation of the nonlinear dispersive free surface flow model leads to a straightforward coupling with the linear seismic wave equations, which are also written in first order hyperbolic form. It furthermore allows the use of explicit time integrators with a rather generous CFL-type time step restriction associated with the dispersive water waves, compared to numerical schemes applied to classical dispersive models. Since the two systems employed are written in the same form of a first order hyperbolic system they can also be efficiently solved in a unique numerical framework. We choose the family of arbitrary high order accurate discontinuous Galerkin finite element schemes. The developed methodology is carefully assessed by first considering several benchmarks for each system separately showing a good agreement with exact and numerical reference solutions. Finally, also coupled test cases are addressed. Throughout this paper we assume the elastic deformations in the solid to be sufficiently small so that their influence on the free surface water waves can be neglected.

中文翻译：

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非线性色散自由表面水波引起的线性地震波模拟的高阶 ADER-DG 方案

在本文中，我们提出了一种统一的、高阶精确的、完全离散的一步 ADER 不连续伽辽金方法，用于模拟由自由表面水波传播产生的海底线性地震波。非线性色散自由表面流的 Serre-Green-Naghdi 模型的双曲线重构与海底线性弹性波传播的一阶速度-应力公式相结合。定义了笛卡尔非一致性网格，并通过弹性波传播的三维域中适当的瞬态压力边界条件来实现耦合，其中压力是水柱中静水压力和非静水压力的组合海底之上。非线性色散自由表面流模型的一阶双曲线重构的使用导致与线性地震波方程的直接耦合，线性地震波方程也以一阶双曲线形式表示。此外，与应用于经典色散模型的数值方案相比，它还允许使用显式时间积分器，其具有与色散水波相关的相当大的 CFL 类型时间步长限制。由于所采用的两个系统是以一阶双曲线系统的相同形式编写的，因此它们也可以在独特的数值框架中有效地求解。我们选择任意高阶精确不连续伽辽金有限元方案族。通过首先分别考虑每个系统的几个基准，对开发的方法进行仔细评估，显示与精确和数值参考解决方案的良好一致性。最后，还解决了耦合测试用例。在整篇论文中，我们假设固体中的弹性变形足够小，以至于它们对自由表面水波的影响可以忽略不计。