Sobre el método de Godunov para sistemas hiperbólicos no conservativos

En este trabajo se aborda la aproximación numérica del problema de Cauchy para sistemas hiperbólicos no conservativos en dimensión uno. El concepto de solución débil de dichos sistemas se define utilizando la teoría de Dal Maso, Le Floch y Murat, basada en la elección de una familia de caminos en el espacio de estados. En primer lugar, establecemos hipótesis para la elección de esta familia de caminos y estudiamos sus implicaciones, en particular, la obtención de una expresión del método de Godunov que generaliza su expresión clásica para sistemas de leyes de conservación. A continuación se estudian las propiedades de buen equilibrado de estos métodos. Finalmente, probamos la consistencia del esquema numérico obtenido con la definición de soluciones débiles. En concreto, probamos que, bajo la hipótesis de variación total acotada, si las aproximaciones obtenidas mediante un método de Godunov basado en una familia de caminos converge uniformemente a alguna función cuando la malla se refina, entonces esta función es una solución débil del sistema no conservativo, relativa a esa familia de caminos. Este resultado se extiende a los esquemas numéricos basados en Resolvedores de Riemann Aproximados

Well-balanced Finite volume solvers

In this work we introduce a general family of finite volume methods for non-homogeneous hyperbolic systems with non-conservative terms. We prove that all of them are “asymptotically well-balanced”: They preserve all smooth stationary solutions in all the domain but a set whose measure tends to zero as ∆x tends to zero. This theory is applied to solve the bilayer Shallow-Water equations with arbitrary cross-section. Finally, some numerical tests are presented for simplified but meaningful geometries, comparing the computed solution with approximated asymptotic analytical solutions

Un esquema de alto orden de tipo MUSTA para la resolución numérica de sistemas hiperbólicos no conservativos

En este trabajo se presenta la extensión de los esquemas MUSTA (Multi-Stage) de alto orden a problemas no conservativos. En [7] E.F. Toro, V.A. Titarev, MUSTA Schemes for Systems of Conservation Laws. J. Comput. Phys., 216(2): 403 - 429, 2006 se presentaron los esquema de tipo MUSTA para leyes de conservación, usando como resolvedor de Riemann aproximado tanto en las etapas de predicción como de corrección el esquema GFORCE. Estos esquemas destacan por su simplicidad y su bajo coste computacional. En este trabajo formulamos el esquema GFORCE en el marco de los esquemas numéricos Ψ-conservativos (“camino-conservativo”) introducidos por Parés en [5] C. Parés. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Num. Anal. 44(1): 300-321, 2006. El esquema MUSTA se reinterpreta como un esquema de reconstrucción de estados, donde el operador de reconstrucción usado está ligado a la resolución de los problemas de Riemann asociados a cada intercelda. Finalmente se propone el uso de un operador de reconstrucción de estados de alto orden, que combinado con la estrategia MUSTA, resulta un esquema de alto orden de tipo MUSTA. Se presentan además algunos ensayos numéricos para el sistema de ecuaciones de aguas someras

Un esquema de volúmenes finitos de alto orden para las ecuaciones de aguas someras con topografía y áreas secas

Presentamos un esquema de volúmenes finitos para la resolución de las ecuaciones de aguas someras con término fuente debido a la topografía del fondo. Se trata de un esquema de alto orden, bien equilibrado y capaz de afrontar situaciones en las que aparecen zonas secas. El esquema se ha desarrollado en un marco no conservativo general, y se basa en reconstrucciones hiperbólicas de estados. El tratamiento de situaciones seco/mojado se lleva a cabo mediante la resolución de problemas de Riemann no lineales en las interceldas donde se detecta una transición

IFCP solver for the the two-layer SWS.

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 method

High-order fully well-balanced numerical methods for one-dimensional blood flow with discontinuous properties

In this paper, we are interested in the numerical study of the one-dimensional blood flow model with discontinuous mechanical and geometrical properties. We present the mathematical model together with its nondimensional form. We do an exhaustive investigation of all its stationary solutions and we propose high-order fully well-balanced numerical methods that are able to preserve all of them. They are based on the combination of the Generalized Hydrostatic Reconstruction and well-balanced reconstruction operators. These methods are able to deal with more than one discontinuous parameter. Several numerical tests are shown to prove its well-balanced and high-order properties, and its convergence to the exact solutions.The research of EPG and CP was partially supported by the Spanish Government (SG), the European Regional Development Fund (ERDF), the Regional Government of Andalusia (RGA), and the University of Málaga (UMA) through the projects of reference RTI2018-096064-B-C21 (SG-ERDF), UMA18-Federja-161 (RGA-ERDF-UMA), and P18-RT-3163 (RGA-ERDF). EPG was also financed by the European Union – NextGenerationEU. // Funding for open access charge: Universidad de Málaga / CBU

On a shallow water model for the simulation of turbidity currents

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

Two-dimensional sediment transport models in shallow water equations. A second order finite volume approach on unstructured meshes

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

Numerical treatment of the loss of hyperbolicity of the two-layer SWS.

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 strateg

Implicit and semi-implicit well-balanced finite-volume methods for systems of balance laws

The aim of this work is to design implicit and semi-implicit high-order well-balanced finite-volume numerical methods for 1D systems of balance laws. The strategy introduced by two of the authors in some previous papers for explicit schemes based on the application of a well-balanced reconstruction operator is applied. The well-balanced property is preserved when quadrature formulas are used to approximate the averages and the integral of the source term in the cells. Concerning the time evolution, this technique is combined with a time discretization method of type RK-IMEX or RK-implicit. The methodology will be applied to several systems of balance laws.This work is partially supported by projects RTI2018-096064-B-C21 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. G.Russo and S.Boscarino acknowledge partial support from the Italian Ministry of University and Research (MIUR), PRIN Project 2017 (No. 2017KKJP4X) entitled “Innovative numerical methods for evolu-tionary partial differential equations and applications”. I. Gómez-Bueno is also supported by a Grant from “El Ministerio de Ciencia, Innovación y Universidades”, Spain (FPU2019/01541) funded by MCIN/AEI/10.13039/501100011033 and “ESF Invest-ing in your future”. // Funding for open access charge: Universidad de Málaga/CBUA