Multilayer shallow water models with locally variable number of layers and semi-implicit time discretization

We propose an extension of the discretization approaches for multilayer shallow water models, aimed at making them more flexible and efficient for realistic applications to coastal flows. A novel discretization approach is proposed, in which the number of vertical layers and their distribution are allowed to change in different regions of the computational domain. Furthermore, semi-implicit schemes are employed for the time discretization, leading to a significant efficiency improvement for subcritical regimes. We show that, in the typical regimes in which the application of multilayer shallow water models is justified, the resulting discretization does not introduce any major spurious feature and allows again to reduce substantially the computational cost in areas with complex bathymetry. As an example of the potential of the proposed technique, an application to a sediment transport problem is presented, showing a remarkable improvement with respect to standard discretization approaches

Flexible and efficient discretizations of multilayer models with variable density

We show that the semi-implicit time discretization approaches previously introduced for multilayer shallow water models for the barotropic case can be also applied to the variable density case with Boussinesq approximation. Furthermore, also for the variable density equations, a variable number of layers can be used, so as to achieve greater flexibility and efficiency of the resulting multilayer approach. An analysis of the linearized system, which allows to derive linear stability parameters in simple configurations, and the resulting spatially semi-discretized equations are presented. A number of numerical experiments demonstrate the effectiveness of the proposed approach

Multilayer methods for geophysical flows: modelling and numerical approximation.

Esta tesis se enmarca en el ámbito de la Matemática Aplicada y la Mecánica de Fluidos Computacional. Concretamente, aborda el modelado matemático y la simulación numérica de flujos geofísicos mediante modelos multicapa. Las contribuciones principales se encuentran en los Capítulos 2, 3 y 4. En el Capítulo 1 se revisa brevemente la aproximación multicapa para las ecuaciones de Navier-Stokes con viscosidad constante, así como el procedimiento para obtener un modelo multicapa. Las avalanchas granulares se han estudiado principalmente mediante modelos integrados. Sin embargo, esos modelos no reproducen variaciones en tiempo de los per les de velocidad. En el Capítulo 2 se presenta un modelo multicapa para avalanchas granulares secas considerando una viscosidad variable de nida por la ley constitutiva (I). En este modelo no se prescribe el per l normal de velocidad horizontal, lo que permite reproducir fuertes cambios en tiempo de estos per les. En el Capítulo 3 se extiende el modelo multicapa anterior al caso de una masa granular con nada en un canal rectangular, para lo que se añade un nuevo término de fricción en las paredes laterales. Se presenta también un esquema numérico bien equilibrado para este modelo, con un tratamiento espec co de los términos correspondientes a la fricción y la reologa. Se muestra que el término de fricción lateral modi ca signi cativamente la evolución de la avalancha. En particular, altera completamente el per l vertical de velocidad, dando lugar a zonas donde el material queda estático bajo una capa superior que se mueve. As mismo, se prueba que incluir el término de fricción lateral en modelos integrados de una capa puede dar lugar a soluciones carentes de sentido desde el punto de vista físico. En el Capítulo 4 se presenta una discretización semi-implícita en tiempo para modelos multicapa, para los que se obtiene una condición CFL menos restrictiva en el caso de un flujo subcrítico, lo que permite reducir notablemente el coste computacional. La descripción multicapa propuesta es novedosa, ya que el número de capas verticales puede cambiar a lo largo del dominio computacional, sin una pérdida de precisión relevante. Estas técnicas se aplican a problemas de flujos oceánicos y de transporte de sedimento

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

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.This research has been partially supported by the Spanish Government and FEDER through the research projects RTI2018-096064-B-C2(1/2) and PID2020-114688RB-I00, the Junta de Andalucía research project P18-RT-3163, the Junta de Andalucia-FEDER-University of Málaga research project UMA18-FEDERJA-16. Funding for open access charge: Universidad de Málaga / CBUA

A flexible <i>z</i>-layers approach for the accurate representation of free surface flows in a coastal ocean model (SHYFEM v. 7_5_71)

We propose a discrete multilayer shallow water model based on z-layers, which, thanks to the insertion and removal of surface layers, can deal with an arbitrarily large tidal oscillation independently of the vertical resolution. The algorithm is based on a classical two-step procedure used in numerical simulations with moving boundaries (grid movement followed by a grid topology change, that is, the insertion/removal of surface layers), which avoids the appearance of surface layers with very small or negative thickness. With ad hoc treatment of advection terms at nonconformal edges that may appear owing to insertion/removal operations, mass conservation and the compatibility of the tracer equation with the continuity equation are preserved at a discrete level. This algorithm called z-surface-adaptive, can be reduced, as a particular case when all layers are moving, to the z-star coordinate. With idealized and realistic numerical experiments, we compare the z-surface-adaptive against z-star and we show that it can be used to simulate coastal flows effectively.</p

A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers

We present a novel staggered semi-implicit hybrid FV/FE method for the numerical solution of the shallow water equations at all Froude numbers on unstructured meshes. A semi-discretization in time of the conservative Saint-Venant equations with bottom friction terms leads to its decomposition into a first order hyperbolic subsystem containing the nonlinear convective term and a second order wave equation for the pressure. For the spatial discretization of the free surface elevation an unstructured mesh of triangular simplex elements is considered, whereas a dual grid of the edge-type is employed for the computation of the depth-averaged momentum vector. The first stage of the proposed algorithm consists in the solution of the nonlinear convective subsystem using an explicit Godunov-type FV method on the staggered grid. Next, a classical continuous FE scheme provides the free surface elevation at the vertex of the primal mesh. The semi-implicit strategy followed circumvents the contribution of the surface wave celerity to the CFL-type time step restriction making the proposed algorithm well-suited for low Froude number flows. The conservative formulation of the governing equations also allows the discretization of high Froude number flows with shock waves. As such, the new hybrid FV/FE scheme is able to deal simultaneously with both, subcritical as well as supercritical flows. Besides, the algorithm is well balanced by construction. The accuracy of the overall methodology is studied numerically and the C-property is proven theoretically and validated via numerical experiments. The solution of several Riemann problems attests the robustness of the new method to deal also with flows containing bores and discontinuities. Finally, a 3D dam break problem over a dry bottom is studied and our numerical results are successfully compared with numerical reference solutions and experimental data

An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

In this work, we present a novel numerical discretization of a variable pressure multilayer shallow water model. The model can be written as a hyperbolic PDE system and allows the simulation of density driven gravity currents in a shallow water framework. The proposed discretization consists in an unlimited arbitrary high order accurate (ADER) Discontinuous Galerkin (DG) method, which is then limited with the MOOD paradigm using an a posteriori subcell finite volume limiter. The resulting numerical scheme is arbitrary high order accurate in space and time for smooth solutions and does not destroy the natural subcell resolution inherent in the DG methods in the presence of strong gradients or discontinuities. A numerical strategy to preserve non-trivial stationary solutions is also discussed. The final method is very accurate in smooth regions even using coarse or very coarse meshes, as shown in the numerical simulations presented here. Finally, a comparison with a laboratory test, where empirical data are available, is also performed.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Funding for open access charge: Universidad de Málaga / CBU