High-Order Unstructured Lagrangian One-Step WENO Finite Volume Schemes for Non-Conservative Hyperbolic Systems: Applications to Compressible Multi-Phase Flows

In this article we present the first better than second order accurate unstructured Lagrangian-type one-step WENO finite volume scheme for the solution of hyperbolic partial differential equations with non-conservative products. The method achieves high order of accuracy in space together with essentially non-oscillatory behavior using a nonlinear WENO reconstruction operator on unstructured triangular meshes. High order accuracy in time is obtained via a local Lagrangian space-time Galerkin predictor method that evolves the spatial reconstruction polynomials in time within each element. The final one-step finite volume scheme is derived by integration over a moving space-time control volume, where the non-conservative products are treated by a path-conservative approach that defines the jump terms on the element boundaries. The entire method is formulated as an Arbitrary-Lagrangian-Eulerian (ALE) method, where the mesh velocity can be chosen independently of the fluid velocity. The new scheme is applied to the full seven-equation Baer-Nunziato model of compressible multi-phase flows in two space dimensions. The use of a Lagrangian approach allows an excellent resolution of the solid contact and the resolution of jumps in the volume fraction. The high order of accuracy of the scheme in space and time is confirmed via a numerical convergence study. Finally, the proposed method is also applied to a reduced version of the compressible Baer-Nunziato model for the simulation of free surface water waves in moving domains. In particular, the phenomenon of sloshing is studied in a moving water tank and comparisons with experimental data are provided

Central Schemes for Nonconservative Hyperbolic Systems

In this work we present a new approach to the construction of high order finite volume central schemes on staggered grids for general hyperbolic systems, including those not admitting a conservation form. The method is based on finite volume space discretization on staggered cells, central Runge-Kutta time discretization, and integration over a family of paths, associated to the system itself, for the generalization of the method to nonconservative systems. Applications to the one and the two layers shallow water models as prototypes of systems of balance laws and systems with source terms and nonconservative products respectively, will be illustrated

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

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

High order extensions of Roe schemes for two dimensional nonconservative hyperbolic systems

This paper is concerned with the development of well-balanced high order Roe methods for two-dimensional nonconservative hyperbolic systems. In particular, we are interested in extending the methods introduced in [3] to the two-dimensional case. We also investigate the well-balance properties and the consistency of the resulting schemes. We focus in applications to one and two layer shallow water system