This paper introduces a reduced-order modeling approach based on finite volume methods for hyperbolic systems, combining Proper Orthogonal Decomposition (POD) with the Discrete Empirical Interpolation Method (DEIM) and Proper Interval Decomposition (PID). Applied to systems such as the transport equation with source term, non-homogeneous Burgers equation, and shallow water equations with non-flat bathymetry and Manning friction, this method achieves significant improvements in computational efficiency and accuracy compared to previous time-averaging techniques. A theoretical result justifying the use of well-balanced Full-Order Models (FOMs) is presented. Numerical experiments validate the approach, demonstrating its accuracy and efficiency. Furthermore, the question of prediction of solutions for systems that depend on some physical parameters is also addressed, and a sensitivity analysis on POD parameters confirms the model's robustness and efficiency in this case.

Granular avalanches that enter a body of water present a hazard to coastal communities via their ability to generate tsunami waves. By improving the accuracy of model estimates of tsunami severity and quantifying the error of said models, we can improve the accuracy of tsunami hazard assessments. Sensitivity analyses were performed of the hybrid finite-difference finite-element model HySEA to changes in the number of vertical layers used to discretize the water column (from a single layer up to 5 layers), non-hydrostatic vs hydrostatic fluid pressures, different friction rheological laws (Pouliquen and μ(I)), and various friction coefficients. Flume experiments with matching conditions were used to benchmark simulations. An improved fit between simulation and experiment wave heights and wave frequencies, as well as improved model stability was found with 3 to 5 vertical water discretization layers compared to using a single water layer. We also demonstrate an improved fit between simulation and experiment wave heights, speeds, and form with non-hydrostatic versus hydrostatic conditions, although most simulations performed here did not accurately estimate wave speeds. Only minor changes in fit were observed between the Pouliquen and μ(I) friction laws. We demonstrate that multilayer HySEA can reliably estimate tsunami wave heights with a reasonable certainty (< 38% error) without any fitting across different granular mass volumes, grain sizes, and slopes of flow and can reach errors < 2% with only limited fitting. Furthermore, we detail how to improve model fit using additional rheological information (e.g., grain size, slope, volume of mass) and friction coefficients.

In the pioneer work of Savage and Hutter [7] the first depth-averaged model for granular landslides was proposed written in local coordinates over a reference plane. From then, there has been proposed a hugh amount of variations, corrections and generalizations. In particular, for their impact in real applications, we mention the work developed in [1] to include the topography curvature effects and the new friction law proposed in [6] to better reproduce the spread of the granular mass through a velocity dependent friction coefficient. In [3] a correction of the friction law has been proposed in order to improve the motion/stop criterion in terms of the repose angle of the material. This correction have any influence for uniform flows, nevertheless, it plays an important role in case of a free surface not parallel to the reference plane. In this work we show two new applications of this model to relevant situations. In the first one we show the influence of the correction on the spread area of the avalanche over variable topography. In the second one a comparison with experimental data of a submarine avalanche is presented.

Climate change is increasingly predisposing polar regions to large landslides. Tsunamigenic landslides have occurred recently in Greenland ( Kalaallit Nunaat ), but none have been reported from the eastern fjords. In September 2023, we detected the start of a 9-day-long, global 10.88-millihertz (92-second) monochromatic very-long-period (VLP) seismic signal, originating from East Greenland. In this study, we demonstrate how this event started with a glacial thinning–induced rock-ice avalanche of 25 × 10 6 cubic meters plunging into Dickson Fjord, triggering a 200-meter-high tsunami. Simulations show that the tsunami stabilized into a 7-meter-high long-duration seiche with a frequency (11.45 millihertz) and slow amplitude decay that were nearly identical to the seismic signal. An oscillating, fjord-transverse single force with a maximum amplitude of 5 × 10 11 newtons reproduced the seismic amplitudes and their radiation pattern relative to the fjord, demonstrating how a seiche directly caused the 9-day-long seismic signal. Our findings highlight how climate change is causing cascading, hazardous feedbacks between the cryosphere, hydrosphere, and lithosphere.

We investigate the dynamics and deposits of granular flows and the amplitude of landslide-generated water waves using the HySEA depth-averaged shallow-water numerical model, both at laboratory and field scales. We evaluate the different sources of error by quantitatively comparing the simulations with (i) new laboratory experiments of granular collapses in different conditions (dry, immersed, dry flow entering water) and slope angles and (ii) numerical simulations made with the SHALTOP code that describes topography effects better than most depth-averaged landslide-tsunami models. For laboratory configurations, representing the limits of the shallow-water approximation in such models, we show that topography and non-hydrostatic effects are crucial. When topography effects are accounted for empirically-by artificially increasing the friction coefficient and performing non-hydrostatic simulations-the model is able to reproduce the granular mass deposit and the waves recorded at gauges located at a distance of more than two to three times the characteristic dimension of the slide with an error ranging from 1 to 25 per cent depending on the scenario, without any further calibration. Taking into account this error estimate, we simulate landslides that occurred on Montagne Pelee volcano, Martinique, Lesser Antilles as well as the generated waves. Multiple collapse simulations support the assumption that large flank collapses on Montagne Pelee likely occurred in several successive subevents. This result has a strong impact on the amplitude of the generated waves and thus on the associated hazards. In the context of the ongoing seismic volcanic unrest at Montagne Pelee volcano, we calculate the debris avalanche and associated tsunamis for two potential flank-collapse scenarios.

In this work, we present a family of layer-averaged models for the Navier–Stokes equations. For its derivation, we consider a layerwise linear vertical profile for the horizontal velocity component. As a particular case, we also obtain layer-averaged models with the common layerwise constant approximation of the horizontal velocity. The approximation of the derivatives of the velocity components is set by following the theory of distributions to account for the discontinuities at the internal interfaces. Several models has been proposed, depending on the order of approximation of an asymptotic analysis respect to the shallowness parameter. Then, we obtain a hydrostatic model with vertical viscous effects, a hydrostatic model where the pressure depends on the stress tensor, and fully non-hydrostatic models, with a complex rheology. It is remarkable that the proposed models generalize plenty of previous models in the literature. Furthermore, all of them satisfy an exact dissipative energy balance. We also propose a model that is second-order accurate in the vertical direction thanks to a correction of the shear stress approximation. Finally, we show how effective the layerwise linear approach is to notably improve, with respect to the layerwise constant method, the approximation of the velocity profile for some geophysical flows. Namely, a Newtonian fluid and some complex viscoplastic (dry granular and Herschel–Bulkley) materials are considered.

This paper focus on the numerical approximation of two-layer shallow water system. First, a new approximation of the eigenvalues of the system is proposed, which satisfies some interesting properties. From this approximation, we give an accurate estimation of the non-hyperbolic region, which improves significantly the one computed with the classic eigenvalues approximation. In particular, we estimate both the lower and upper boundaries of the non-hyperbolic region. We also give a simple algorithm that allows us to compute bounds for the external eigenvalues, even when complex eigenvalues arise. We design an efficient FV solver depending on the hyperbolic nature of the system, that combines the PVM-2U-FL scheme and a new solver introduced in this work, named IFCP-FL, which degenerates to the Lax–Wendroff method in smooth areas. Different strategies for the numerical treatment for the loss of hyperbolicity are considered and discussed. Two of them are based on adding a friction term, depending on the classic or the new eigenvalues approximations, and the third one considers a simplified hyperbolic model in areas with complex eigenvalues. Some numerical tests are performed, including the case of two-layer fluids with different ratio of densities. The application to a two-layer model for submarine landslides is also considered. In the latter case, we show how the treatments based on adding friction are not appropriate for this kind of applications, whereas the treatment based on changing to a hyperbolic model produces much better results.

Starting from Navier-Stokes' equation we derive two shallow water multilayer models for yield stress fluids, depending on the asymptotic analysis. One of them takes into account the normal stress contributions, making possible to recover a pseudoplug layer instead of a purely plug zone. A specific numerical scheme is designed to solve this model thanks to a finite volume discretization. It involves well-balancing techniques to be able to compute accurately the transitions between yielded and unyielded (or pseudoplug) zones, an important feature of the original partial differential equations' model. We perform numerical simulations on various test cases relevant to these physics: analytical solution of a uniform flow, steady solutions for arrested state, and a viscoplastic dam break. Simulations agree well when we perform comparisons with physical experiments of the group of Christophe Ancey (EPFL) and we make a comparative study including shallow water models and lubrication models that they present in Ancey et al. (2012) [3]. Thanks to the multilayer structure of our model, we can go further on the description of the vertical structure associated to the (bottom) sheared layer and the top (pseudo-)plug layer.

A new family of non-hydrostatic layer-averaged models for the non-stationary Euler equations is presented in this work, with improved dispersion relations. They are a generalisation of the layer-averaged models introduced in Fernández-Nieto et al. (Commun Math Sci 16(05):1169–1202, 2018), named LDNH models, where the vertical profile of the horizontal velocity is layerwise constant. This assumption implies that solutions of LDNH can be seen as a first order Galerkin approximation of Euler system. Nevertheless, it is not a fully (x, z) Galerkin discretisation of Euler system, but just in the vertical direction (z). Thus, the resulting model only depends on the horizontal space variable (x), and therefore specific and efficient numerical methods can be applied (see Escalante-Sanchez et al. in J Sci Comput 89(55):1–35, 2021). This work focuses on particular weak solutions where the horizontal velocity is layerwise linear on z and possibly discontinuous across layer interfaces. This approach allows the system to be a second-order approximation in the vertical direction of Euler system. Several closure relations of the layer-averaged system with non-hydrostatic pressure are presented. The resulting models are named LIN-NH _k models, with k=0,1,2 . Parameter k indicates the degree of the vertical velocity profile considered in the approximation of the vertical momentum equation. All the introduced models satisfy a dissipative energy balance. Finally, an analysis and a comparison of the dispersive properties of each model are carried out. We show that Models LIN-NH _1 and LIN-NH _2 provide a better dispersion relation, group velocity and shoaling than LDNH models.

Since May 2018, Mayotte island has experienced an important seismic activity linked to the on-going sismo-volcanic crisis. Although variations in the number of earthquakes and in their distribution have been observed since the start of the eruption in early July 2018, a continuous seismicity persists. It could weaken the steep submarine slopes of Mayotte, as highlighted by the high-resolution bathymetry data collected during the MAYOBS cruise in May 2019. This could trigger submarine landslides with associated tsunamis.To address the hazards associated with such events, we analyzed geomorphological data to define 8 scenarios of potential submarine landslides with volumes ranging from 11,25.106 to 800.106 m3. We simulated the resulting landslide dynamics as well as generated waves (Poulain et al. 2022). In order to estimate the uncertainty associated to the modeling approach, a hierarchy of different model approximations was tested, spanning hydrostatic, non-hydrostatic and multilayer approaches. A sensitivity analysis was also performed by varying the initial released mass, the rheological parameters describing the landslide, its interaction with the water column, the Manning friction coefficient as well as the resolution of the bathymetry description. The combination of all these elements provides an estimate of the uncertainty on simulation results. We show that, in the context of Mayotte, non-hydrostatic effects have the most prominent influence on simulated water elevation and waves velocity. Other key factors include the friction coefficient within the landslide and the resolution of the bathymetry. These results show that landslide-tsunami models should still be improved as well as the estimates of the parameters involved to reduce the related uncertainties on the water wave calculation (water elevation, velocity) that can exceed a factor two.Poulain, P., et al. (2022). Numerical simulation of submarine landslides and generated tsunamis: application to the on-going Mayotte seismo-volcanic crisis. Comptes Rendus. Géoscience, 354(S2), 1-30.

The multilayer model proposed in this paper is a generalization of the multilayer non-hydrostatic model for shallow granular flows (Fernández-Nieto et al in Commun Math Sci 16(5):1169–1202, 2018. https://doi.org/10.4310/cms.2018.v16.n5.a1 ), the multilayer model with μ (I) rheology (Fernández-Nieto et al in J Fluid Mech 798:643–681, 2016. https://doi.org/10.1017/jfm.2016.333 ), and the monolayer model with weakly non-hydrostatic pressure for dry granular flows (Garres-Díaz et al in J Sci Comput, 2021. https://doi.org/10.1007/s10915-020-01377-9 ). We show that the proposed model verifies a dissipative energy balance. A well-balanced numerical scheme is proposed to solve the equations based on a projection method and a hydrostatic reconstruction for the Coulomb friction terms. In order to reduce the computational cost associated with solving the linear system of the projection method, a precomputing of the initial guess for an iterative solver is proposed. This strategy allows us to reduce the computational time by around 70 % when 20 layers are considered. In the numerical tests, we show that the proposed model can recover the in-depth velocity profiles typically observed in lab experiments and capture the flow/no-flow interface that appears in granular avalanches. During the initial stage of granular collapse simulations, the model is shown to improve the approximation of the mass profiles compared to other models and to predict the parabolic shape of the front velocity evolution with time, as observed in lab experiments. Interestingly, our numerical tests show that the ability of the granular flow to overcome obstacles strongly depends on the model used, which is of strong interest for landslide hazard assessment.

In this work efficient semi-implicit methods for sediment bedload transport models with gravitational effects under subcritical regimes is proposed. Several families of models with gravitational effects are presented and rewritten under a general formulation that allows us to apply the semi-implicit method. In the numerical tests we focus on the application of a generalization of the Ashida-Michiue model, which includes the gradient of both the bedload and the fluid surface. Analytical steady states solutions (both lake at rest and non vanishing velocity) are deduced an approximated with the proposed scheme. In all the pre-sented tests, the computational efforts are notably reduced thanks to the proposed method without losing the accuracy in the results.(c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

Datasets of the 6 presented experiments with 4mm beads presented in the paper "Performance and limits of a shallow-water model for landslide-generated tsunamis: from laboratory experiments to simulations of flank collapses at Montagne Pelée (Martinique)". Each datasets (csv file) corresponds to the hand picked profile of either the water free surface or the granular material at 0.1 second of interval. The third dataset for each experiment corespond of the gauges records.

Depth-averaged models, such as the Savage-Hutter model with Coulomb or Pouliquen friction laws, are usually considered to simulate aerial and submarine avalanches. In particular, submarine avalanches can be the source of a tsunami. These models are presented in local coordinates over the topography or a reference bottom. We show in this work that classical models do not in some cases preserve the physical threshold of motion. On the one hand, the simulated granular mass can start to flow even if the slope angle of its free surface is lower than the repose angle of the granular material involved. On the other hand, the granular mass can stay at rest being the slope angle of the free surface higher than the repose angle of the material. Several numerical tests are presented to illustrate these problems related to classical depth averaged models. In this work we also propose an initial correction which ensures that the model preserves, up to the second order, the physical threshold of motion defined by the repose angle of the material. Several numerical tests are presented, by comparing also with experimental data to illustrate the effect of the proposed correction.

Depth-averaged models, such as the Savage-Hutter model with Coulomb or Pouliquen friction laws, do not in some cases preserve the physical threshold of motion. In particular, the simulated granular mass can start to flow (or stay at rest) even if the slope angle of its free surface is lower (or higher) than the repose angle of the granular material involved. The problem is related to the hydrostatic pressure assumption, associated with the direction of integration, which is orthogonal to a reference plane or a reference bottom. We propose here an initial method to correct this misleading behavior. Firstly, we define a correction of the friction term that accounts for the Jacobian of a change of coordinates, making it possible to reproduce the physical threshold of motion and thus the solutions at rest. Secondly, we observe that the 3D model presented in [F. Bouchut, I. Ionescu, and A. Mangeney. An analytic approach for the evolution of the static-flowing interface in viscoplastic granular flows. Commun, Math. Sci. , 14(8):2101-2126, 2016] verifies the physical thresholds of motion because it is based on a second order correction of the pressure valid for slow granular flows. The correction proposed here ensures that the model preserves, up to the second order, the physical threshold of motion defined by the repose angle of the material. Several numerical tests are presented to illustrate certain problems related to classical depth averaged models and the remedial effect of the proposed correction, in particular through comparisons with experimental data. We finally show that this correction is not exact far from the starting and stopping phases of the granular avalanche and should be improved by adding other second order terms in the pressure approximation. (c) 2022 The Author(s). Published by Elsevier Inc. ( http://creativecommons.org/licenses/by/4.0/ )

Since May 2018, Mayotte island has experienced an important seismic activity linked to the on-going sismo-volcanic crisis. The epicenters of the seismic swarms are located between 5 and 15 km east of Petite Terre for the main swarm, and 25 km east of Petite Terre for the secondary swarm. Although variations in the number of earthquakes and their distribution have been observed since the start of the eruption in early July 2018 [Lemoine A.(2020), Cesca et al.(2020)], a continuous seismicity persists and could generate several earthquakes of magnitudes close to M4 widely felt by the population. This recurrent seismicity could weaken the steep submarine slopes of Mayotte, as highlighted by the high resolution bathymetry data collected during the MAYOBS cruise in May 2019 (Feuillet et al.,submitted) and trigger submarine landslides with associated tsunamis. To address the hazards associated with such events, we analyzed morphological data to define 8 scenarios of potential submarine slides with volumes ranging from 11,25.106 to 800.106 m3 and we simulate the landslide dynamics and generated waves. We use two complementary numerical models: (i) the code HYSEA to simulate the dynamic of the submarine granular flows and the water wave generation, and (ii) the Boussinesq FUNWAVE- TVD model simulate the waves propagation and the inundation on Mayotte. The effect of the time at which the models are coupled is investigated. The most impacting submarine slide scenarios are located close to Petite Terre at a shallow depth. They can locally generate a sea surface elevation more than a meter in local areas especially at Petite Terre. The various simulations show that parts of the island are particularly sensitive to the risk of tsunamis. Indeed, some scenarios that does not cause significant coastal flooding still seems to cause significant hazards in these exposed areas. The barrier reef around Mayotte has a prominent role in controlling the wave propagation towards the island and therefore reducing the impact on land. It should be noted that the arrival of tsunamis on the coastline is not necessarily preceded by a retreat from the sea and the waves can reach the coasts of Mayotte very quicky (few minutes). Cesca, S., Letort, J., Razafindrakoto, H.N.T. et al. Drainage of a deep magma reservoir near Mayotte inferred from seismicity and deformation. Nat. Geosci. 13, 87–93 (2020). https://doi.org/10.1038/s41561-019-0505-5 Feuillet, N, Jorry, S. J., Crawford, W, Deplus, C. Thinon, I, Jacques, E. Saurel, J.M., Lemoine, A., Paquet, F., Daniel, R., Gaillot, A., Satriano, C., Peltier, A., Aiken, C., Foix, O., Kowalski, P., Laurent, A., Beauducel, F., Grandin, R., Ballu, V., Bernard, P., Donval, J.P., Geli, L., Gomez, J. Guyader, V., Pelleau, P., Rinnert, E., Bertil, D., Lemarchand, A., Van der Woerd, J.et al. (in rev). Birth of a large volcano offshore Mayotte through lithosphere-scale rifting, Nature. Anne Lemoine, Pierre Briole, Didier Bertil, Agathe Roullé, Michael Foumelis, Isabelle Thinon, Daniel Raucoules, Marcello de Michele, Pierre Valty, Roser Hoste Colomer, The 2018–2019 seismo-volcanic crisis east of Mayotte, Comoros islands: seismicity and ground deformation markers of an exceptional submarine eruption, Geophysical Journal International, Volume 223, Issue 1, October 2020, Pages 22–44, https://doi.org/10.1093/gji/ggaa273

We investigate the dynamics and deposits of granular flows and the amplitude of the generated water waves using the depth-averaged shallow numerical model HySEA, both at the lab- and field scales. We investigate the different sources of errors by quantitatively comparing the simulations with (i) six new laboratory experiments of granular collapses in different conditions (dry, immersed, dry flow entering water) and slope angles, and (ii) numerical simulations made with the code SHALTOP that describes topography effects better than most landslide-tsunami models. In the laboratory configurations, at the limit of the shallow-approximation in such models, we show that topography and non-hydrostatic effects are crucial. However, when empirically accounting for topography effects by artificially increasing the friction coefficient and performing non-hydrostatic simulations, the model is able to reproduce the granular mass deposit and the waves recorded at gauges located at a distance of more than 2-3 times the characteristic dimension of the slide, with an error ranging from 1 % to 25 % depending on the scenario, without any further calibration. Taking into account this error estimation, we simulate landslides that occurred on Montagne Pelée volcano, Martinique, Petites Antilles as well as the generated waves. Results support the hypothesis that large flank collapse events in Montagne Pelée likely occurred in several successive sub-events. This result has a strong impact on the amplitude of the generated waves, and thus on the associated hazards. In the context of the on-going seismic volcanic unrest at Montagne Pelée volcano, we calculate the debris avalanche and associated tsunami for two potential flank-collapse scenarios.

A two-layer shallow water type model is proposed to describe bedload sediment transport for strong and weak interactions between the fluid and the sediment. The key point falls into the definition of the friction law between the two layers, which is a generalization of those introduced in Fernández-Nieto et al. ( https://doi.org/10.1051/m2an/2016018 ). Moreover, we prove formally that the two-layer model converges to a Saint-Venant-Exner system (SVE) including gravitational effects when the ratio between the hydrodynamic and morphodynamic time scales is small. The SVE with gravitational effects is a degenerated nonlinear parabolic system, whose numerical approximation can be very expensive from a computational point of view, see for example T. Morales de Luna et al. ( https://doi.org/10.1007/s10915-010-9447-1 ). In this work, gravitational effects are introduced into the two-layer system without any parabolic term, so the proposed model may be a advantageous solution to solve bedload sediment transport problems.