Accurate simulation of shallow flows using arbitrary order ADER schemes and overcoming numerical shockwave anomalies

En la actualidad, gracias al desarrollo de algoritmos de simulación avanzados y de tecnologías computacionales eficientes que ha tenido lugar durante las últimas décadas, es posible simular problemas de elevada complejidad que hace unos años eran inalcanzables. Parte de estos problemas se modelan mediente ecuaciones en derivadas parciales de tipo hiperbólico. Este tipo de ecuaciones reproducen con fidelidad aquellos fenómenos que involucran la propagación de ondas. En situaciones realistas, es necesario tener en cuenta efectos dinámicos adicionales más allá de los fenómenos puramente convectivos. Dichos efectos se modelan matemáticamente mediante los llamados términos fuente, que dan lugar a sistemas de ecuaciones no homogéneos y suponen un desafío computacional importante en numerosas ocasiones. Sólo unas determinadas discretizaciones del término fuente garantizan la convergencia de la solución a una solución físicamente realista; cuando se utilizan métodos numéricos sofisticados, la complejidad en el tratamiento de los términos fuentes aumenta de forma notable.Esta tesis se centra en el desarrollo de esquemas numéricos de orden arbitrario para la resolución de sistemas hiperbólicos siguiendo la metodología ADER, que permite la extensión del esquema tradicional de Godunov a orden arbitrario. Los métodos que aquí se presentan están enfocados a la resolución de las ecuaciones de aguas poco profundas, pero se formulan de forma general para su posible aplicación a otros modelos matemáticos. La particularidad fundamental de los esquemas numéricos propuestos en esta tesis reside en la manera en la que se introducen los términos fuente en la formulación discreta. A diferencia de la mayoría de métodos comunmente utilizados, aquí se propone introducir los términos fuente en la formulación de los flujos numéricos, siguiendo una metodología de discretización upwind. Esto implica considerar los términos fuente en la formulación del problema de Riemann derivativo. De este modo, es posible garantizar un equilibrio perfecto entre flujos y términos fuente a nivel discreto y reproducir con precisión aquellas situaciones de equilibrio relevantes para los problemas estudiados. Para las ecuaciones de aguas poco profundas, aquellos esquemas que satisfacen esta propiedad se denominaron tradicionalmente well-balanced, aunque dicha atribución sólo hacía referencia a la preservación de estados de reposo estático.Se muestra que sólo aquellos términos fuentes de tipo geométrico (por ejemplo, término de variación de fondo en las ecuaciones de aguas poco profundas) se deben incluir en la resolución del problema de Riemann derivativo. Otros términos fuente de distinta naturaleza se pueden integrar de forma tradicional utilizando reglas de cuadratura, o bien, se pueden reescribir como términos geométricos y pueden ser tratados del mismo modo. Siguiendo esta última aproximación, es posible garantizar la propiedad well-balanced sin perder el orden de convergencia arbitrario. Aquí se detalla la construcción de esquemas numéricos de orden arbitrario para las ecuaciones de aguas poco profundas con términos fuente de fondo, fricción y Coriolis, que satisfacen la propiedad well-balanced. Además, mediante consideraciones de conservación de energía a nivel discreto, dicha propiedad se extiende para situaciones de equilibrio unidimensionales que involucran velocidades no nulas, desde una perspectiva de un esquema ADER.Por último, en este trabajo también se estudian anomalías numéricas que pueden aparecer en la resolución de las ecuaciones de aguas poco profundas. Dichas anomalías son intrínsecas al método de volúmenes finitos y pueden dar lugar a oscilaciones severas de la solución numérica. Siguiendo estudios previos sobre anomalías numéricas en las ecuaciones de Euler, se formula un marco teórico para el estudio de dichas anomalías en las ecuaciones de aguas poco profundas. Se muestra que la presencia de resaltos hidráulicos genera oscilaciones numéricas en el caudal y se propone una corrección del flujo que lo solventa.<br /

Asymptotically and exactly energy balanced augmented flux-ADER schemes with application to hyperbolic conservation laws with geometric source terms

In this work, an arbitrary order HLL-type numerical scheme is constructed using the flux-ADER methodology. The proposed scheme is based on an augmented Derivative Riemann solver that was used for the first time in Navas-Montilla and Murillo (2015) 1]. Such solver, hereafter referred to as Flux-Source (FS) solver, was conceived as a high order extension of the augmented Roe solver and led to the generation of a novel numerical scheme called AR-ADER scheme. Here, we provide a general definition of the FS solver independently of the Riemann solver used in it. Moreover, a simplified version of the solver, referred to as Linearized-Flux-Source (LFS) solver, is presented. This novel version of the FS solver allows to compute the solution without requiring reconstruction of derivatives of the fluxes, nevertheless some drawbacks are evidenced. In contrast to other previously defined Derivative Riemann solvers, the proposed FS and LFS solvers take into account the presence of the source term in the resolution of the Derivative Riemann Problem (DRP), which is of particular interest when dealing with geometric source terms. When applied to the shallow water equations, the proposed HLLS-ADER and AR-ADER schemes can be constructed to fulfill the exactly well-balanced property, showing that an arbitrary quadrature of the integral of the source inside the cell does not ensure energy balanced solutions. As a result of this work, energy balanced flux-ADER schemes that provide the exact solution for steady cases and that converge to the exact solution with arbitrary order for transient cases are constructed

the WAF method for non-homogeneous SWE with pollutant

This paper deals with the extension of the WAF method to discretize Shallow Water Equations with pollutants. We consider two different versions of the WAF method, by approximating the intermediate waves using the flux of HLL or the direct approach of HLLC solver. It is seen that both versions can be written under the same form with different definitions for the approximation of the velocity waves. We also propose an extension of the method to non-homogeneous systems. In the case of homogeneous systems it is seen that we can rewrite the third component of the numerical flux in terms of an intermediate wave speed approximation. We conclude that – in order to have the same relation for non-homogeneous systems – the approximation of the intermediate wave speed must be modified. The proposed extension of the WAF method preserves all stationary solutions, up to second order accuracy, and water at rest in an exact way, even with arbitrary pollutant concentration. Finally, we perform several numerical tests, by comparing it with HLLC solver, reference solutions and analytical solutions

Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations

After we derive the Serre system of equations of water wave theory from a generalized variational principle, we present some of its structural properties. We also propose a robust and accurate finite volume scheme to solve these equations in one horizontal dimension. The numerical discretization is validated by comparisons with analytical, experimental data or other numerical solutions obtained by a highly accurate pseudo-spectral method.Comment: 28 pages, 16 figures, 75 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

A comprehensive explanation and exercise of the source terms in hyperbolic systems using Roe type solutions. Application to the 1D-2D shallow water equations

Powerful numerical methods have to consider the presence of source terms of different nature, that intensely compete among them and may lead to strong spatiotemporal variations in the flow. When applied to shallow flows, numerical preservation of quiescent equilibrium, also known as the well-balanced property, is still nowadays the keystone for the formulation of novel numerical schemes. But this condition turns completely insufficient when applied to problems of practical interest. Energy balanced methods (E-schemes) can overcome all type of situations in shallow flows, not only under arbitrary geometries, but also with independence of the rheological shear stress model selected. They must be able to handle correctly transient problems including modeling of starting and stopping flow conditions in debris flow and other flows with a non-Newtonian rheological behavior. The numerical solver presented here satisfies these properties and is based on an approximate solution defined in a previous work. Given the relevant capabilities of this weak solution, it is fully theoretically derived here for a general set of equations. This useful step allows providing for the first time an E-scheme, where the set of source terms is fully exercised under any flow condition involving high slopes and arbitrary shear stress. With the proposed solver, a Roe type first order scheme in time and space, positivity conditions are explored under a general framework and numerical simulations can be accurately performed recovering an appropriate selection of the time step, allowed by a detailed analysis of the approximate solver. The use of case-dependent threshold values is unnecessary and exact mass conservation is preserved

2D Finite Volume Numerical Modeling of Free Surface Flows with Topography

In this thesis a finite-volume MUSCL-type scheme for the numerical solution of inhomogeneous SWE is presented. The novel aspect is data reconstruction: the scheme, named WSDGM (Weighted Surface-Depth Gradient Method), computes intercell water depths performing a weighted average of DGM and SGM reconstructions, in which the weight function depends on the local Froude number. This combination makes WSDGM capable of performing a robust tracking of wet/dry fronts and, together with an unsplit centered discretization of the bed slope source term, of exactly maintaining the static condition on non-flat topographies (C-property). Moreover, a numerical procedure performing a correction of the numerical fluxes in the computational cells with water depth smaller than a fixed tolerance enables a drastic reduction of the mass error in the presence of wetting and drying fronts. The effectiveness and robustness of the proposed scheme were assessed by comparing numerical results with the analytical and reference solutions of a set of test cases. Finally, to check the numerical model to field-scale applications, the results of two hypothetical dam-break events are reported