Multidimensional approximate Riemann solvers for hyperbolic systems

Esta tesis doctoral se centra en el desarrollo de resolvedores de Riemann multidimensionales incompletos eficientes para sistemas hiperbólicos generales, aplicables tanto en el caso conservativo como en el no conservativo. Dichos resolvedores se construyen a partir de un modelo de cuatro ondas, dadas por las velocidades de propagación maximales en cada vértice de una malla estructurada. En particular, se construye una versión simple de un esquema HLL 2D bien equilibrado, la cual se toma como base para diseñar una clase más general de resolvedores de Riemann incompletos 2D, los denominados esquemas AVM (Approximate Viscosity Matrix). La gran ventaja de los esquemas AVM es la posibilidad de controlar la cantidad de difusión numérica considerada para cada sistema hiperbólico, con un coste computacional razonable. Se demuestra que los esquemas numéricos de primer orden resultantes son consistentes con el sistema hiperbólico considerado, y linealmente estables bajo una condición CFL de hasta la unidad. Tales esquemas pueden ser usados como base para construir esquemas de alto orden. En esta tesis, se construye un esquema de segundo orden mediante el método predictor-corrector MUSCL-Hancock. Para analizar las propiedades de los esquemas propuestos, se han considerado experimentos numéricos en magnetohidrodinámica (MHD) y sistemas de aguas someras (SWE) de una y dos capas. En el caso de MHD, la condición de divergencia nula se ha impuesto mediante una nueva técnica basada en la escritura no conservativa de las ecuaciones. Por otro lado, para SWE, la presencia de la topografía del fondo y de los términos de acoplamiento entre capas representan una dificultad adicional, que se resuelve dentro del marco de los esquemas camino-conservativos. Por último, se ha desarrollado un algoritmo simple y eficiente para la implementación de los esquemas en tarjetas gráficas (GPU), con el objetivo de aumentar la eficiencia computacional

Solving shallow-water systems in 2D domains using Finite Volume methods and multimedia SSE instructions

AbstractThe goal of this paper is to construct efficient parallel solvers for 2D hyperbolic systems of conservation laws with source terms and nonconservative products. The method of lines is applied: at every intercell a projected Riemann problem along the normal direction is considered which is discretized by means of well-balanced Roe methods. The resulting 2D numerical scheme is explicit and first-order accurate. In [M.J. Castro, J.A. García, J.M. González, C. Pares, A parallel 2D Finite Volume scheme for solving systems of balance laws with nonconservative products: Application to shallow flows, Comput. Methods Appl. Mech. Engrg. 196 (2006) 2788–2815] a domain decomposition method was used to parallelize the resulting numerical scheme, which was implemented in a PC cluster by means of MPI techniques.In this paper, in order to optimize the computations, a new parallelization of SIMD type is performed at each MPI thread, by means of SSE (“Streaming SIMD Extensions”), which are present in common processors. More specifically, as the most costly part of the calculations performed at each processor consists of a huge number of small matrix and vector computations, we use the Intel© Integrated Performance Primitives small matrix library. To make easy the use of this library, which is implemented using assembler and SSE instructions, we have developed a C++ wrapper of this library in an efficient way. Some numerical tests were carried out to validate the performance of the C++ small matrix wrapper. The specific application of the scheme to one-layer Shallow-Water systems has been implemented on a PC’s cluster. The correct behavior of the one-layer model is assessed using laboratory data

A consistent intermediate wave speed for a well-balanced HLLC solver.

In this work, we present a HLLC scheme modification for application to nonhomogeneous shallow-water equations with pollutant transport. This new version is related to the definition of a consistent approximation of the intermediate wave speed. Numerical results are presented to illustrate the importance of such approximation to get appropriate pollutant concentration profiles

Parallelization of a relaxation scheme modelling the bedload transport of sediments in shallow water flow

In this work we are interested in numerical simulations for bedload erosion processes. We present a relaxation solver that we apply to moving dunes test cases in one and two dimensions. In particular we retrieve the so-called anti-dune process that is well described in the experiments. In order to be able to run 2D test cases with reasonable CPU time, we also describe and apply a parallelization procedure by using domain decomposition based on the classical MPI library.Comment: 19 page

A robust well-balanced scheme for multi-layer shallow water equations

International audienceThe numerical resolution of the multi-layer shallow water system encounters two additional difficulties with respect to the one-layer system. The first is that the system involves nonconservative terms, and the second is that it is not always hyperbolic. A splitting scheme has been proposed by Bouchut and Morales, that enables to ensure a discrete entropy inequality and the well-balanced property, without any theoretical difficulty related to the loss of hyperbolicity. However, this scheme has been shown to often give wrong solutions. We introduce here a variant of the splitting scheme, that has the overall property of being conservative in the total momentum. It is based on a source-centered hydrostatic scheme for the one-layer shallow water system, a variant of the hydrostatic scheme. The final method enables to treat an arbitrary number of layers, with arbitrary densities and arbitrary topography. It has no restriction concerning complex eigenvalues, it is well-balanced and it is able to treat vacuum, it satisfies a semi-discrete entropy inequality. The scheme is fast to execute, as is the one-layer hydrostatic method