Introducción al Método de Volúmenes finitos

Se presentará una introducción a las leyes de conservación escalares, incluyendo el cálculo de la solución de problemas de Riemann. Se presentarán los métodos de tipo Godunov y se discutirá la estabilidad en el caso lineal. Se comentará la extensión de los métodos a alto orden usando operadores de reconstrucción.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

A Subsonic-Well-Balanced Reconstruction Scheme for Shallow Water Flows

International audienceWe consider the Saint-Venant system for shallow water flows with nonflat bottom. In past years, efficient well-balanced methods have been proposed in order to well resolve solutions close to steady states at rest. Here we describe a strategy based on a local subsonic steady state reconstruction that allows one to derive a subsonic-well-balanced scheme, preserving exactly all the subsonic steady states. It generalizes the now well-known hydrostatic solver, and like the latter it preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality. An application to the Euler-Poisson system is proposed

A weakly non-hydrostatic shallow model for dry granular flows

A non-hydrostatic depth-averaged model for dry granular flows is proposed, taking into account vertical acceleration. A variable friction coefficient based on the $\mu(I)$ rheology is considered. The model is obtained from an asymptotic analysis in a local reference system, where the non-hydrostatic contribution is supposed to be small compared to the hydrostatic one. The non-hydrostatic counterpart of the pressure may be written as the sum of two terms: one corresponding to the stress tensor and the other to the vertical acceleration. The model introduced here is weakly non-hydrostatic, in the sense that the non-hydrostatic contribution related to the stress tensor is not taken into account due to its complex implementation. A simple and efficient numerical scheme is proposed. It consists of a three-step splitting procedure, and it is based on a hydrostatic reconstruction. Two key points are: (i) the friction force has to be taken into account before solving the non-hydrostatic pressure. Otherwise, the incompressibility condition is not ensured; (ii) both the hydrostatic and the non-hydrostatic pressure are taken into account when dealing with the friction force. The model and numerical scheme are then validated based on several numerical tests, including laboratory experiments of granular collapse. The influence of non-hydrostatic terms and of the choice of the coordinate system (Cartesian or local) is analyzed. We show that non-hydrostatic models are less sensitive to the choice of the coordinate system. In general, the non-hydrostatic model introduced here much better reproduces granular collapse experiments compared to hydrostatic models. An important result is that the simulated mass profiles up to the deposit and the front velocity are greatly improved. As expected, the influence of the non-hydrostatic pressure is shown to be larger for small values of the slope

Implicit and implicit-explicit Lagrange-projection finite volume schemes exactly well-balanced for 1D shallow water system

In this paper we consider the Lagrange-Projection technique in the framework of finite volume schemes applied to the shallow water system. We shall consider two versions of the scheme for the Lagrangian step: one fully implicit and one implicit-explicit, based on how the geometric source term is treated. First and second order well-balanced versions of the schemes are presented, in which the water at rest solutions are preserved. This allows to obtain efficient numerical schemes in low Froude number regimes, as the usual CFL restriction driven by the acoustic waves is avoided.This work is partially supported by projects RTI2018-096064-B-C21 and RTI2018-096064-B-C22 funded by MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe”, projects P18-RT-3163 of Junta de Andalucía and UMA18-FEDERJA-161 of Junta de Andalucía-FEDER-University of Málaga. C. Caballero-Cárdenas is supported by the grant FPI2019/087773 funded by MCIN/AEI/10.13039/501100011033 and “ESF Investing in your future”. // Funding for open access charge: Universidad de Málaga/CBUA

On the influence of the thickness of the sediment moving layer in the definition of the bedload transport formula in Exner systems

In this paper we study Exner system and introduce a modified general definition for bedload transport flux. The new formulation has the advantage of taking into account the thickness of the sediment layer which avoids mass conservation problems in certain situations. Moreover, it reduces to a classical solid transport discharge formula in the case of quasi-uniform regime. We also present several numerical tests where we compare the proposed sediment transport formula with the classical formulation and we show the behavior of the new model in different configurations

Formal deduction of the Saint-Venant-Exner model including arbitrarily sloping sediment beds and associated energy

In this work we present a deduction of the Saint-Venant-Exner model through an asymptotic analysis of the Navier-Stokes equations. A multi-scale analysis is performed in order to take into account that the velocity of the sediment layer is smaller than the one of the uid layer. This leads us to consider a shallow water type system for the uid layer and a lubrication Reynolds equation for the sediment one. This deduction provides some improvements with respect to the classical Saint- Venant-Exner model: (i) the deduced model has an associated energy. Moreover, it allows us to explain why classical models do not have an associated energy and how to modify them in order to recover a model with this property. (ii) The model incorporates naturally a necessary modi cation that must be taken into account in order to be applied to arbitrarily sloping beds. Furthermore, we show that this modi cation is di erent of the ones considered classically, and that it coincides with a classical one only if the solution has a constant free surface. (iii) The deduced solid transport discharge naturally depends on the thickness of the moving sediment layer, what allows to ensure sediment mass conservation. Moreover, we include a simpli ed version of the model for the case of quasi-stationary regimes. Some of these simpli ed models correspond to the generalization of classical ones such as Meyer- Peter&M uller and Ashida-Michiue models. Three numerical tests are presented to study the evolution of a dune for several de nition of the repose angle, to see the in uence of the proposed de nition of the e ective shear stress in comparison with the classical one, and by comparing with experimental data

A multilayer shallow water system for polydisperse sedimentation

This work considers the flow of a fluid containing one disperse substance consisting of small particles that belong to different species differing in size and density. The flow is modelled by combining a multilayer shallow water approach with a polydisperse sedimentation process. This technique allows one to keep information on the vertical distribution of the solid particles in the mixture, and thereby to model the segregation of the particle species from each other, and from the fluid, taking place in the vertical direction of the gravity body force only. This polydisperse sedimentation process is described by the well-known Masliyah-Lockett-Bassoon (MLB) velocity functions. The resulting multilayer sedimentation-ow model can be written as a hyperbolic system with nonconservative products. The definitions of the nonconservative products are related to the hydrostatic pressure and to the mass and momentum hydrodynamic transfer terms between the layers. For the numerical discretization a strategy of two steps is proposed, where the first one is also divided into two parts. In the _rst step, instead of approximating the complete model, we approximate a reduced model with a smaller number of unknowns. Then, taking advantage of the fact that the concentrations are passive scalars in the system, we approximate the concentrations of the different species by an upwind scheme related to the numerical flux of the total concentration. In the second step, the effect of the transference terms defined in terms of the MLB model is introduced. These transfer terms are approximated by using a numerical ux function used to discretize the 1D vertical polydisperse model (see Bürger, García, Karlsen and Towers, J. Eng. Math. 60 (2008), 387{425). Finally, some numerical examples are presented. Numerical results suggest that the multilayer shallow water model could be adequate in situations where the settling takes place from a suspension that undergoes horizontal movement

Shallow Water Moment models for bedload transport problems

In this work a simple but accurate shallow model for bedload sediment transport is proposed. The model is based on applying the moment approach to the Shallow Water Exner model, making it possible to recover the vertical structure of the flow. This approach allows us to obtain a better approximation of the fluid velocity close to the bottom, which is the relevant velocity for the sediment transport. A general Shallow Water Exner moment model allowing for polynomial velocity profiles of arbitrary order is obtained. A regularization ensures hyperbolicity and easy computation of the eigenvalues. The system is solved by means of an adapted IFCP scheme proposed here. The improvement of this IFCP type scheme is based on the approximation of the eigenvalue associated to the sediment transport. Numerical tests are presented which deal with large and short time scales. The proposed model allows to obtain the vertical structure of the fluid, which results in a better description on the bedload transport of the sediment layer

A general vertical decomposition of Euler equations: Multilayer-moment models

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

An Efficient Two-Layer Non-hydrostatic Approach for Dispersive Water Waves

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