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[Abstract]: The goal of this paper is to develop 2nd order Implicit-Explicit Runge-Kutta (IMEX-RK) finite volume (FV) schemes for solving 1d parabolic PDEs for option pricing, with possible nonlinearities in the source and advection terms. The spatial semi-discretization of the advection is carried out by combining finite volume methods with 2nd order state reconstructions; while the diffusive terms are discretized using second order finite differences. The time integration is performed by means of IMEX-RK time integrators: the advection is treated explicitly, and the diffusion, implicitly. The obtained numerical schemes have several advantages: they are computationally very efficient, thanks to the implicit discretization of the diffusion in the IMEX-RK time integrators, that allows to overcome the strict time step restriction; they yield second order accuracy for even nonlinear problems and with non-regular initial conditions; and they can be extended to higher order.

[Abstract]: We present a novel and general methodology for building second order finite volume implicit-explicit Runge-Kutta numerical schemes for solving two dimensional financial parabolic PDEs with mixed derivatives. The methods achieve second order convergence even in the presence of non-regular initial conditions. The IMEX time integrator allows to overcome the tiny time-step induced by the diffusive term in the explicit schemes, also providing accurate and non-oscillatory approximations of the Greeks.

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.

In this work, we present a general framework for vertical discretizations of Euler equations. It generalizes the usual moment and multilayer models and allows to obtain a family of multilayer-moment models. It considers a multilayer-type discretization where the layerwise velocity is a polynomial of arbitrary degree N on the vertical variable. The contribution of this work is twofold. First, we compare the multilayer and moment models in their usual formulation, pointing out some advantages/disadvantages of each approach. Second, a family of multilayer-moment models is proposed. As particular interesting case we shall consider a multilayer-moment model with layerwise linear horizontal velocity. Several numerical tests are presented, devoted to the comparison of multilayer and moment methods, and also showing that the proposed method with layerwise linear velocity allows us to obtain second order accuracy in the vertical direction. We show as well that the proposed approach allows to correctly represent the vertical structure of the solutions of the hydrostatic Euler equations. Moreover, the measured efficiency shows that in many situations, the proposed multilayer-moment model needs just a few layers to improve the results of the usual multilayer model with a high number of vertical layers.

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 Pelée volcano, Martinique, Lesser Antilles as well as the generated waves. Multiple collapse simulations support the assumption that large flank collapses on Montagne Pelée 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 Pelée volcano, we calculate the debris avalanche and associated tsunamis for two potential flank-collapse scenarios.

Since May 2018, Mayotte Island has been experiencing seismo-volcanic activities that could trigger submarine landslides and, in turn, tsunamis. To address these hazards, we use the HySEA numerical model to simulate granular flow dynamics and the Boussinesq FUNWAVE-TVD numerical model to simulate wave propagation and subsequent inundations. We investigate 8 landslide scenarios (volumes from 11.25 × 106 m3 to 800 × 106 m3 ). The scenario posing the greatest threat involves destabilization on the eastern side of Mayotte’s lagoon at a shallow depth and can generate sea-surface deformations of up to 2 m. We show that the barrier reef surrounding Mayotte plays a prominent role in controlling water-wave propagation and in protecting the island. The tsunami travel time to the coast is very short (a few minutes) and the tsunami is not necessarily preceded by a sea withdrawal. Our simulation results provide a key to establishing hazard maps and evacuation plans and improving early-warning systems., Depuis mai 2018, l’île de Mayotte connaît une activité sismo-volcanique importante susceptible de déclencher des glissements de terrain sous-marins générant des tsunamis. Pour faire face à ces aléas, nous utilisons deux modèles numériques complémentaires : le modèle HySEA (simulant la dynamique des écoulements granulaires) et le modèle Boussinesq FUNWAVE-TVD (simulant la propagation des vagues et les inondations) pour étudier 8 scénarios de glissements sous-marins potentiels (volumes de 11,25 × 106 m3 à 800 × 106 m3). Les scénarios ayant le plus d’impact se situent à proximité de Petite Terre et à faible profondeur. Ils peuvent générer une élévation de la surface de la mer jusqu’à 2 m en zone habitée à Petite Terre. Nous montrons que la barrière de corail entourant Mayotte joue un rôle prépondérant dans le contrôle de la propagation des vagues et dans la protection de l’île. Le temps de trajet du tsunami jusqu’à la côte est très court (quelques minutes) et le tsunami n’est pas nécessairement précédé d’un retrait maritime. De telles observations sont essentielles pour construire des cartes d’aléas précises et des plans d’évacuation afin d’aider la population.

[Abstract]: The goal of this article is to make automatic data assimilation for a landslide tsunami model, given by the coupling between a non-hydrostatic multi-layer shallow-water and a Savage-Hutter granular landslide model for submarine avalanches. The coupled model is discretized using a positivity preserving second-order path-conservative finite volume scheme. Then, the data assimilation problem is posed in a global optimization framework. Later, multi-path parallel metaheuristic stochastic global optimization algorithms are developed. More precisely, a multi-path Simulated Annealing algorithm is compared with a multi-path hybrid global optimization algorithm based on coupling Simulated Annealing with gradient local searchers.

In this paper, we propose a two-layer depth-integrated non-hydrostatic system with improved dispersion relations. This improvement is obtained through three free parameters: two of them related to the representation of the pressure at the interface and a third one that controls the relative position of the interface concerning the total height. These parameters are then optimized to improve the dispersive properties of the resulting system. The optimized model shows good linear wave characteristics up to kH ≈ 10, that can be improved for long waves. The system is solved using an efficient formally second-order well-balanced and positive preserving hybrid finite volume/difference numerical scheme. The scheme consists of a two-step algorithm based on a projection-correction type scheme. First, the hyperbolic part of the system is discretized using a Polynomial Viscosity Matrix path-conservative finite-volume method. Second, the dispersive terms are solved using finite differences. The method has been applied to idealized and challenging physical situations that involve nearshore breaking. Agreement with laboratory data is excellent. This technique results in an accurate and efficient method.

We propose a multi-layer approach to simulate hyperpycnal and hypopycnal plumes in flows with free surface. The model allows to compute the vertical profile of the horizontal and the vertical components of the velocity of the fluid flow. The model can describe as well the vertical profile of the sediment concentration and the velocity components of each one of the sediment species that form the turbidity current. To do so, it takes into account the settling velocity of the particles and their interaction with the fluid. This allows to better describe the phenomena than a single layer approach. It is in better agreement with the physics of the problem and gives promising results. The numerical simulation is carried out by rewriting the multi-layer approach in a compact formulation, which corresponds to a system with non-conservative products, and using path-conservative numerical scheme. Numerical results are presented in order to show the potential of the model.

A physical-biological coupled model was used to estimate the effect of the physical processes at the Strait of Gibraltar over the biogeochemical features of the Atlantic Inflow (AI) towards the Mediterranean Sea. This work was focused on the seasonal variation of the biogeochemical patterns in the AI and the role of the Strait; including primary production and phytoplankton features. As the physical model is 1D (horizontal) and two-layer, different integration methods for the primary production in the Biogeochemical Fluxes Model (BFM) have been evaluated. An approach based on the integration of a production-irradiance function was the chosen method. Using this Plankton Functional Type model (BFM), a simplified phytoplankton seasonal cycle in the AI was simulated. Main results included a principal bloom in spring dominated by nanoflagellates, whereas minimum biomass (mostly picophytoplankton) was simulated during summer. Physical processes occurring in the Strait could trigger primary production and raise phytoplankton biomass (during spring and autumn), mainly due to two combined effects. First, in the Strait a strong interfacial mixing (causing nutrient supply to the upper layer) is produced, and, second, a shoaling of the surface Atlantic layer occurs eastward. Our results show that these phenomena caused an integrated production of 105 g C m− 2 year− 1 in the eastern side of the Strait, and would also modify the proportion of the different phytoplankton groups. Nanoflagellates were favored during spring/autumn while picophytoplankton is more abundant in summer. Finally, AI could represent a relevant source of nutrients and biomass to Alboran Sea, fertilizing the upper layer of this area with 4.95 megatons nitrate year− 1 (79.83 gigamol year− 1) and 0.44 megatons C year− 1. A main advantage of this coupled model is the capability of solving relevant high-resolution processes as the tidal forcing without expensive computing requirements, allowing to assess the effe

The key goals to studying submarine landslides and associated tsunamis are to better assess the magnitude information of palaeo-tsunamis and to contribute to assessing future tsunami risks. The numerical modelling of submarine landslides and the tsunamis thus generated comprises important interdisciplinary research that requires knowledge of both geology and numerical modelling. Models are capable of delineating the time evolution of tsunami hydrodynamics, sediment transport, and the resulting morphological changes associated with deposition. To advance towards the ultimate goal of improved tsunami risk assessment, modellers and geologists need to develop an in-depth mutual understanding of the advantages, limitations, and uncertainties in both numerical modelling and geological records. In this work, seafloor features related to former submarine landslides in the Alboran Sea basin have been identified through multibeam bathymetry data and high- to very high-resolution seismic profiles. Themathematicalmodelling and hindcasting have been performed through numerical simulation of a hypothetical submarine landslide and the associated tsunami that could have originated one of the submarine landslides identified. This system, on the southern Alboran Ridge slope, is currently reworked by turbidite processes, forming a submarine canyon-sedimentary fan system known as the Al-Borani System. The HySEA numerical model simulates the submarine landslide triggering the tsunami and the water mass evolution, wave propagation, and the final penetration of the tsunami waves onto the coast, reproducing initial and subsequent tsunami wave impacts by means of a single coupled numerical model. This numerical model allows an analysis of the influence of basin morphology on the tsunami propagation features, such as shape and propagation patterns, speed or wave amplitude and, finally, its impact on the coast (in this case South Iberia and North Africa). This model can also be used as a pred

In this work we propose a second order flux limiter finite volume method, named PVM-2U-FL, that only uses information of the two external waves of the hyperbolic system. This method could be seen as a natural extension of the well known WAF method introduced by Prof. Toro in [21]. We prove that independently of the number of unknowns of the 1D system, it recovers the second order accuracy at regular zones, while in presence of discontinuities, the scheme degenerates to PVM-2U method, which can be seen as an improvement of the HLL method (see [4], [8]). Another interesting property of the method is that it does not need any spectral decomposition of the Jacobian or Roe matrix associated to the flux function. Therefore, it can be easily applied to systems with a large number of unknowns or in situations where no analytical expression of the eigenvalues or eigenvectors are known. In this work, we apply the proposed method to Magnetohydrodynamics and to stratified multilayer flows. Comparison with the twowaves WAF and HLL-MUSCL methods are also presented. The numerical results show that PVM-2U-FL is the most efficient and accurate among them.

Los modelos NPZ (siglas para Nutrientes-Fitoplancton-Zooplancton en ingl es) son com unmente utilizados en estudios de biolog a marina. Este tipo de modelos utiliza un conjunto de ecuaciones diferenciales muy sencillo para de nir la din amica del plancton marino. Las variables de estado de las cuales se modeliza su evoluci on son los nutrientes, el toplancton y el zooplankton, esto se hace en t erminos de su contenido de nitr ogeno, ya que este compuesto el que normalmente limita la producci on primaria en el oc eano. En este trabajo se implementa el acoplado de un modelo NPZ para el modelado de los ujos biogeoqu micos con un modelo de aguas poco profundas bicapa para la hidrodin amica. El objetivo es aplicar este modelo a la simulaci on de ujos biogeoqu micos en el Estrecho de Gibraltar con imposici on de la din amica mareal.

Los modelos NPZ (siglas para Nutrientes-Fitoplancton-Zooplancton en inglés) son comúnmente utilizados en estudios de biología marina. Este tipo de modelos utiliza un conjunto de ecuaciones diferenciales muy sencillo para medir la dinámica del plancton marino. Las variables de estado de las cuales se modeliza su evolución son los nutrientes, el toplancton y el zooplankton, esto se hace en términos de su contenido de nitrógeno, ya que este compuesto el que normalmente limita la producción primaria en el océano. En este trabajo se implementa el acoplado de un modelo NPZ para el modelado de los flujos biogeoquímicos con un modelo de aguas poco profundas bicapa para la hidrodinámica. El objetivo es aplicar este modelo a la simulación de flujos biogeoquímicos en el Estrecho de Gibraltar con imposición de la dinámica mareal.

This paper focuses on the generalization of the HLLC Riemann solver for nonconservative problems. First, the general ideas of the extension of the HLLC solvers for nonconservative systems are discussed. Then, two particular HLLC solvers are described for a turbidity current model with sediment transport. Some results concerning the positivity of the corresponding schemes are presented. Several numerical tests are performed to compare the two HLLC solvers among them and with a Roe method.

The numerical solutions of shallow water equations are useful for applications related to geophysical flows that usually take place in large computational domains and could require real time calculation. Therefore, parallel versions of accurate and efficient numerical solvers for high performance platforms are needed to be able to deal with these simulation scenarios in reasonable times. In this paper we present an efficient CUDA implementation of a first and second order HLL methods and a two-waves TVD-WAF one. We propose to write all these methods under a common framework, such as, their CUDA implementations share the same structure. In particular, the reformulation of TVD-WAF numerical flux and the improved definition of the flux limiter allows us to obtain a more robust solver in situations like wet/dry fronts. Finally, some numerical tests are presented showing that the TVD-WAF method is slightly slower that the first order HLL method and two times faster than the second order HLL method, but it provides numerical results almost as accurate as the second order HLL scheme.

In this work, we present a class of fast first order finite volume solvers, named as PVM (Polynomial Viscosity Matrix), for balance laws or, more generally, for nonconservative hyperbolic systems. They are defined in terms of viscosity matrices computed by a suitable polynomial evaluation of a Roe matrix. These methods have the advantage that they only need some information about the eigenvalues of the system to be defined, and no spectral decomposition of Roe Matrix is needed. As consequence, they are faster than Roe method. These methods can be seen as a generalization of the schemes introduced by Degond et al. in [12] for balance laws and nonconservative systems. The first-order path conservative methods to be designed here are intended to be used as the basis for higher order methods for multi-dimensional problems. In this work, some well known solvers as Rusanov, Lax-Friedrichs, FORCE (see [30], [8]), GFORCE (see [31], [8]) or HLL (see [18]) are redefined under this form, and then some new solvers are proposed. Finally, some numerical tests are presented and the performance of the numerical schemes are compared among them and with Roe scheme

The numerical solution of two-layer shallow water systems is required to simulate accurately stratified fluids, which are ubiquitous in nature: they appear in atmospheric flows, ocean currents, oil spills, . . . Moreover, the implementation of the numerical schemes to solve these models in realistic scenarios imposes huge demands of computing power. In this paper, we tackle the acceleration of these simulations in triangular meshes by exploiting the combined power of several CUDA-enabled GPUs in a GPU cluster. For that purpose, an improvement of a path conservative Roe type finite volume scheme which is specially suitable for GPU implementation is presented, and a distributed implementation of this scheme which uses CUDA and MPI to exploit the potential of a GPU cluster is developed. This implementation overlaps MPI communication with CPU-GPU memory transfers and GPU computation to increase efficiency. Several numerical experiments performed on a cluster of modern CUDA-enabled GPUs show the efficiency of the distributed solver.

Los componentes principales de este libro son: a) conocer y estudiar determinados fenómenos naturales de consecuencias catastróficas, como los tsunamis, y b) identificar los útiles científicos para representar estos fenómenos. Por ello, es necesario disponer de un profundo conocimiento del fondo marino donde se produce el fenómeno, conocimiento que proviene del análisis de datos geológicos y geofísicos, y por otro lado, es encesario hacer uso de la potencialidad de los modelos matemáticos que permiten realizar simulaciones numéricas precisas. El libro presenta los resultados de la simulación numérica de un tsunami generado en la cuenca del Mar de Alborán y su propagación e interacción con la morfología submarina de dicho área geográfica.

The goal of this article is to design a new approximate Riemann solver for the two-layer shallow water system which is fast compared to Roe schemes and accurate compared to Lax-Friedrichs, FORCE, or GFORCE schemes (see[14]). This Riemann solver is based on a suitable decomposition of a Roe matrix (see [27]) by means of a parabolic viscosity matrix (see [16]) that captures some information concerning the intermediate characteristic fields. The corresponding first order numerical scheme, which is called IFCP (Intermediate Field Capturing Parabola) is linearly L∞-stable, well-balanced, and it doesn’t require an entropy-fix technique. Some numerical experiments are presented to compare the behavior of this new scheme with Roe and GFORCE methods

In this work, a characterization of the hyperbolicity region for the two layer shallow-water system is proposed and checked. Next, some path-conservative finite volume schemes (see [11]) that can be used even if the system is not hyperbolic are presented, but they are not in general L2 linearly stable in that case. Then, we introduce a simple but efficient strategy to enforce the hyperbolicity of the two-layer shallow-water system consisting in adding to the system an extra amount of friction at every cell in which complex eigenvalues are detected at a given time step. The implementation is performed by a predictor/corrector strategy: first a numerical scheme is applied to the unmodified two-layer system, regardless of the hyperbolic character of the system. Next, we check if the predicted cell averages are in the hyperbolic region or not. If not, the mass-fluxes are corrected by adding a quadratic friction law between layers whose coefficient is computed so that the corrected cell average is as near as possible of the boundary of the hyperbolicity region. Finally, some numerical test have been performed to assess the efficiency of the proposed strategy

This paper is concerned with the development of well-balanced high order Roe methods for two-dimensional nonconservative hyperbolic systems. In particular, we are interested in extending the methods introduced in [3] to the two-dimensional case. We also investigate the well-balance properties and the consistency of the resulting schemes. We focus in applications to one and two layer shallow water systems

We present a model for hyperpycnal plumes or turbidity currents that takes into account the interaction between the turbidity current and the bottom, considering deposition and erosion effects as well as solid transport of particles at the bed load due to the current. Water entrainment from the ambient water in which the turbidity current plunges is also considered. Motion of ambient water is neglected and the rigid lid assumption is considered. The model is obtained as a depth-average system of equations under the shallow water hypothesis describing the balance of fluid mass, sediment mass and mean ow. The character of the system is analyzed and numerical simulations are carried out using finite volume schemes and path-conservative Roe schemes.

In this paper, we study the numerical approximation of bedload sediment transport due to shallow layer flows. The hydrodynamical component is modeled by a 2D shallow water system and the morphodynamical component by a solid transport discharge formula that depends on the hydrodynamical variables. The coupled system can be written as a nonconservative hyperbolic system. To discretize it, first we consider a Roe-type first order scheme as well as a variant based on the use of flux limiters. These first order schemes are then extended to second order accuracy by means of a new MUSCL-type reconstruction operator on unstructured meshes. Finally, some numerical tests are presented.

In this paper, we introduce a class of well-balanced finite volume schemes for 2D non-homogeneous hyperbolic systems. We extend the derivation of standard finite volume solvers for homogeneous systems to non-homogeneous ones using the method of lines. We study conservation and some well-balanced properties of the numerical scheme. We apply our solvers to shallow water equations: we prove that these exactly compute the water at rest solutions. We also perform some numerical tests, by comparing with 1D solutions, simulating the formation of a hydraulic drop and a hydraulic jump, and studying a real dam break: Aznalcóllar, an ecological disaster happened in the province of Seville, Spain in 1998.

This paper is concerned with the numerical approximation of bedload sediment transport due to water evolution. For the hydrodynamical component we consider Shallow Water equations. The morphodynamical component is defined by a continuity equation, which is defined in function of the solid transport discharge. We present several deterministic models, such as Meyer-Peter & Müller, Van Rijn or Grass model. We also present an unified definition for the solid transport discharge, and we compare with Grass model. Both components define a coupled system of equations that can be rewrite as a non-conservative hyperbolic system. To discretize it, we consider finite volume methods with or without flux limiters and high order state reconstructions. Finally we present several tests, where we observe numerically the order of the numerical schemes. Comparisons with analytical solutions and experimental data are also presented.

In this paper we present a new two-layer model of Savage-Hutter type to study submarine avalanches. A layer composed of fluidized granular material is assumed to flow within an upper layer composed of an inviscid fluid (e. g. water). The model is derived in a system of local coordinates following a non-erodible bottom and takes into account its curvature. We prove that the model verifies an entropy inequality, preserves water at rest for a sediment layer and their solutions can be seen as particular solutions of incompressible Euler equations under hydrostatic assumptions. Buoyancy effects and the centripetal acceleration of the grain movement due to the curvature of the bottom are considered in the definition of the Coulomb term. We propose a two-step Roe type solver to discretize the presented model. It exactly preserves water at rest and no movement of the sediment layer, when its angle is smaller than the angle of repose, and up to second order all stationary solutions. Finally, some numerical tests are performed by simulating submarine and sub-aerial avalanches as well as the generated tsunami.

En este trabajo se aborda la aproximación numérica del problema de arrastre de sedimentos causada por la evolución del agua. Para la componente hidrodinámica se consideran las ecuaciones de aguas poco profundas. La componente morfodinámica se define mediante una ecuación de continuidad, dada en función del caudal sólido. Ambas componentes constituyen un sistema acoplado que puede reescribirse como un sistema hiperbólico no conservativo (ver [4] A.M. Ferreiro Ferreiro. Desarrollo de técnicas de post-proceso de flujos hidrodinámicos, modelización de problemas de transporte de sedimentos y simulación numérica mediante técnicas de volúmenes finitos. Tesis Doctoral. Universidad de Sevilla. 2006). Se propone un esquema 2D generalizado de Roe con reconstrucciones de estado para sistemas hiperbólicos no conservativos (ver [4]), mediante esquemas de volúmenes finitos y el método de líneas (ver [5] J.A. García Rodríguez. Paralelización de esquemas de vol´umenes finitos: aplicación a la resolución de sistemas de tipo aguas someras. Tesis Doctoral. Universidad de Málaga. 2005), extendiendo los esquemas de alto orden para el caso 1D propuesto en [2] M.J. Castro, J.M. Gallardo and C. Parés. High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow water systems. Math. Comp., 75: 1103-1134. 2006. La reconstrucción de estado empleada es de tipo MUSCL (ver [1] B. Van Leer. MUSCL. A new approach to numerical gas dynamics. Computing in plasma physics and astrophysics, Max-Planck-Institut fur plama physik.. Carchung, Germany, April 1976), que proporciona orden dos para mallas no estructuradas de volúmenes finitos de tipo arista. Finalmente se presenta un test num´erico en el que se estudia la evolución del ángulo de expansi´on de una monta˜na de arena (ver [4]).

En este trabajo se aplican esquemas numéricos bidimensionales de alto orden (ver [5] A.M. Ferreiro Ferreiro. Desarrollo de técnicas de post-proceso de flujos hidrodinámicos, modelización de problemas de transporte de sedimentos y simulación numérica mediante técnicas de volúmenes finitos. Tesis Doctoral. Universidad de Sevilla. 2006) empleando el método de volúmenes finitos, para la modelización del transporte inerte de una sustancia en un fluido, como por ejemplo, el vertido de contaminantes. El modelo matemático consiste en un sistema acoplado de aguas someras y una ecuación que modela de transporte inerte de una sustancia. Dicho acoplamiento introduce un nuevo campo linealmente degenerado en el sistema. Además, si el vertido ocupa sólo una porción de fluido, la frontera del mismo se propaga como una discontinuidad de contacto (ver [11] E. F. Toro. Shock-capturing methods for free-surface shallow flows. Wiley, 2001). En consecuencia, para aproximar con precisión la evolución de un vertido es necesario desarrollar métodos numéricos que capturen adecuadamente las discontinuidades de contacto. En este trabajo presentamos un método de volúmenes finitos de alto orden bidimensional basado en técnicas de reconstrucciones de estados. Concretamente hemos empleado una técnica de tipo MUSCL discontinua que garantiza orden 2. Finalmente, se presentan algunos tests numéricos.

En este trabajo se estudia un nuevo modelo de tipo Savage-Hutter para avalanchas submarinas. El modelo obtenido es de tipo aguas someras bicapa en el que la capa superior representa al fluido y la capa inferior representa a un estrato de material granular. En la deducción del modelo se han tenido en cuenta diferentes leyes constitutivas en los tensores de esfuerzos, la porosidad del estrato de sedimento o roca y un término de fricción de tipo Coulomb. El modelo obtenido puede aplicarse en el estudio de problemas de avalanchas submarinas y algunos tipos de tsunamis. Finalmente se presentan la discretización del modelo obtenido mediante técnicas de vol´umenes finitos bien equilibrados y un ensayo numérico donde se modela una avalancha submarina.