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High order ADER-DG schemes for the simulation of linear seismic waves induced by nonlinear dispersive free-surface water waves. (English) Zbl 1447.74042

Summary: 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. In particular, 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. To this end, Cartesian non-conforming meshes are defined in the solid and fluid domains 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 that contain higher order derivatives and typically require implicit solvers. Since the two systems that describe the seismic waves and the free-surface water waves are written in the same form of a first-order hyperbolic system they can also be efficiently solved in a unique numerical framework. In this paper we choose the family of arbitrary high-order accurate discontinuous Galerkin finite element schemes, which have already shown to be suitable for the numerical simulation of wave propagation problems. The developed methodology is carefully assessed by first considering several benchmarks for each system separately, i.e. in the framework of linear elasticity and non-hydrostatic free-surface flows, 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.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74S20 Finite difference methods applied to problems in solid mechanics
74L05 Geophysical solid mechanics
74J05 Linear waves in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
86A15 Seismology (including tsunami modeling), earthquakes
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References:

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