In some configurations, dispersion effects must be taken into account to improve the simulation of complex fluid flows. A family of free-surface dispersive models has been derived in Fernández-Nieto et al. (Commun Math Sci 16(05):1169–1202, 2018). The hierarchy of models is based on a Galerkin approach and parameterised by the number of discrete layers along the vertical axis. In this paper we propose some numerical schemes designed for these models in a 1D open channel. The cornerstone of this family of models is the Serre – Green-Naghdi model which has been extensively studied in the literature from both theoretical and numerical points of view. More precisely, the goal is to propose a numerical method for the $$LDNH_2$$ L D N H 2 model that is based on a projection method extended from the one-layer case to any number of layers. To do so, the one-layer case is addressed by means of a projection-correction method applied to a non-standard differential operator. A special attention is paid to boundary conditions. This case is extended to several layers thanks to an original relabelling of the unknowns. In the numerical tests we show the convergence of the method and its accuracy compared to the $$LDNH_0$$ L D N H 0 model.

<p>The features of the seismic ruptures, such as the duration of shallow earthquakes in subduction zones, may affect the tsunami generation and the inundation intensity. Numerical and experimental results have shown how the interaction between the shallow part of the fault and the seismic radiation trapped in the hanging wall, can lead to enhanced up-dip rupture propagation. This in turn may result in shallow slip amplification producing larger vertical displacement, and even transient ground motion that is larger than the final static displacement. On the other hand, tsunami modelling for hazard assessment and early warning is generally based on static sea-floor displacement obtained with an instantaneous elastic dislocation (without shallow slip amplification) on a simplified hydrostatic model for tsunami generation and propagation. Here, we aim to analyze the relative importance of these effects and the optimal modelling strategy for the tsunami generation. Using a Tohoku-like setting, we impose time dependent initial conditions as computed from 1-D dynamic rupture simulations, by varying the rupture extent and duration over a wide range of stress-drop, rigidity and average slip values (corresponding to earthquake magnitudes between 7.5 and 9, approximately). We performed 1-D numerical tsunami simulations using both the hydrostatic and the multi-layer non-hydrostatic versions of Tsunami-HySEA. We also account for different coastal morphologies, modelling the presence of shelf and/or fjords and variable slope bathymetry. We address, first, how the time-dependent sea-floor displacement characteristics effects may affect (enhancing or reducing) the tsunamigenic potential. To do this, we investigated the resulting tsunami features, in terms of maximum wave height above sea level (also seaward) and maximum run-up, in relation to the spatial and temporal characteristic scales of the transient sea floor displacement. We also compare the simulations with a time-dependent initial condition against those where a static sea-floor displacement is used. We show that the use of a static source systematically overestimates the tsunami effects on the mainland, with the more realistic tsunami reduced due to the seaward seismic rupture (up-dip) directivity, opposite to the direction of the tsunami propagation. Moreover, the slower the rupture, the larger the overestimation. Conversely, as the rupture slows down, the seismic rupture propagating in the same direction of the tsunami increases the tsunami amplitude toward the open ocean. Second, we wish to assess in which conditions and to what extent it is enough to use a shallow-water tsunami model and when, instead, a more complex tsunami modelling scheme is required. The hydrostatic simulations lead to overestimate the inundation, although less significantly with respect to the static/dynamic comparison. We finally investigate how the discrepancy between simplified and complex modelling is controlled by different trench, shelf, and coastal morphologies.</p>

<p>When tsunamigenic events are simulated in deep to moderately deep waters, frequency dispersion effects may become mandatory. In the framework of dispersive systems, non-hydrostatic pressure type models have been shown to be able to describe weakly dispersive waves [2,3]. Although promising results begin to glimpse nowadays, dispersive solvers are still far from being robust, efficient and able to compute on a faster than real-time (FTRT) basis. The main difficulty that presents this type of systems is that at each time step a parabolic-elliptic problem has to be numerically solved and a high computational effort is required.</p><p>In [1] a novel weakly non-linear and weakly dispersive system that takes into account dispersive effects is presented. The main advantage is that the system is strictly hyperbolic and that any explicit numerical scheme can be applied to solve numerically the equations.</p><p>We will present new numerical results from an upgrade of the system presented in [1], considering curvature effects through a rewriting of the system in spherical coordinates. The numerical results will cover some standard field validation tests involving tsunami propagation waves. Besides, the explicit numerical scheme has been implemented exploiting the power of modern GPU architectures (CUDA). Then, numerical results along with some computational times will show that this numerical model opens a new line on tsunami simulation scenarios, using a new, efficient and accurate procedure to produce FTRT tsunami propagation including dispersive effects.</p><p>Acknowledgments: This research has been partially supported by the Spanish Government Research project MEGAFLOW (RTI2018-096064-B-C21), Universidad de Málaga, Campus de Excelencia Internacional Andalucía Tech and ChEESE project (EU Horizon 2020, grant agreement Nº 823844), https://cheese-coe.eu</p><p>[1] C. Escalante, M. Dumbser, M. Castro, An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution<br>with high order discontinuous galerkin schemes, Journal of Computational Physics 394 (2019) 385 – 416.</p><p>[2] C. Escalante, T. Morales, M. Castro, Non-hydrostatic pressure shallow flows: Gpu implementation using finite volume and finite difference<br>scheme, Applied Mathematics and Computation (2018) 631–659.</p><p>[3] Y. Yamazaki, Z. Kowalik, K. Cheung, Depth-integrated, non-hydrostatic model for wave breaking and run-up, Numerical Methods in Fluids<br>61 (2008) 473–497.</p>

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