Entropy stable DGSEM for nonlinear hyperbolic systems in nonconservative form with application to two-phase flows

In this work, we consider the discretization of nonlinear hyperbolic systems in nonconservative form with the high-order discontinuous Galerkin spectral element method (DGSEM) based on collocation of quadrature and interpolation points (Kopriva and Gassner, J. Sci. Comput., 44 (2010), pp.136--155; Carpenter et al., SIAM J. Sci. Comput., 36 (2014), pp.~B835-B867). We present a general framework for the design of such schemes that satisfy a semi-discrete entropy inequality for a given convex entropy function at any approximation order. The framework is closely related to the one introduced for conservation laws by Chen and Shu (J. Comput. Phys., 345 (2017), pp.~427--461) and relies on the modification of the integral over discretization elements where we replace the physical fluxes by entropy conservative numerical fluxes from Castro et al. (SIAM J. Numer. Anal., 51 (2013), pp.~1371--1391), while entropy stable numerical fluxes are used at element interfaces. Time discretization is performed with strong-stability preserving Runge-Kutta schemes. We use this framework for the discretization of two systems in one space-dimension: a $2\times2$ system with a nonconservative product associated to a linearly-degenerate field for which the DGSEM fails to capture the physically relevant solution, and the isentropic Baer-Nunziato model. For the latter, we derive conditions on the numerical parameters of the discrete scheme to further keep positivity of the partial densities and a maximum principle on the void fractions. Numerical experiments support the conclusions of the present analysis and highlight stability and robustness of the present schemes

Augmented resolution of linear hyperbolic systems under nonconservative form

Hyperbolic systems under nonconservative form arise in numerous applications modeling physical processes, for example from the relaxation of more general equations (e.g. with dissipative terms). This paper reviews an existing class of augmented Roe schemes and discusses their application to linear nonconservative hyperbolic systems with source terms. We extend existing augmented methods by redefining them within a common framework which uses a geometric reinterpretation of source terms. This results in intrinsically well-balanced numerical discretizations. We discuss two equivalent formulations: (1) a nonconservative approach and (2) a conservative reformulation of the problem. The equilibrium properties of the schemes are examined and the conditions for the preservation of the well-balanced property are provided. Transient and steady state test cases for linear acoustics and hyperbolic heat equations are presented. A complete set of benchmark problems with analytical solution, including transient and steady situations with discontinuities in the medium properties, are presented and used to assess the equilibrium properties of the schemes. It is shown that the proposed schemes satisfy the expected equilibrium and convergence properties

Air entrainment in transient flows in closed water pipes: a two-layer approach

In this paper, we first construct a model for free surface flows that takes into account the air entrainment by a system of four partial differential equations. We derive it by taking averaged values of gas and fluid velocities on the cross surface flow in the Euler equations (incompressible for the fluid and compressible for the gas). The obtained system is conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation of this system to finally construct a two-layer kinetic scheme in which a special treatment for the "missing" boundary condition is performed. Several numerical tests on closed water pipes are performed and the impact of the loss of hyperbolicity is discussed and illustrated. Finally, we make a numerical study of the order of the kinetic method in the case where the system is mainly non hyperbolic. This provides a useful stability result when the spatial mesh size goes to zero

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

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

Mathematical and numerical modelling of dispersive water waves

Fecha de lectura de Tesis: 4 diciembre 2018.En esta tesis doctoral se expone en primer lugar una visión general del modelado de ondas dispersivas para la simulación de procesos tsunami-génicos. Se deduce un nuevo sistema bicapa con propiedades de dispersión mejoradas y un nuevo sistema hiperbólico. Además se estudian sus respectivas propiedades dispersivas, estructura espectral y ciertas soluciones analíticas. Así mismo, se ha diseñado un nuevo modelo de viscosidad sencillo para la simulación de los fenómenos físicos relacionados con la ruptura de olas en costa. Se establecen los resultados teóricos requeridos para el diseño de esquemas numéricos de tipo volúmenes finitos y Galerkin discontinuo de alto orden bien equilibrados para sistemas hiperbólicos no conservativos en una y dos dimensiones. Más adelante, los esquemas numéricos propuestos para los sistemas de presión no hidrostática introducidos se describen. Se pueden destacar diferentes enfoques y estrategias. Por un lado, se diseñan esquemas de volúmenes finitos implícitos de tipo proyección-corrección en mallas decaladas y no decaladas. Por otro lado, se propone un esquema numérico de tipo Galerkin discontinuo explícito para el nuevo sistema de EDPs hiperbólico propuesto. Para permitir simulaciones en tiempo real, una implementación eficiente en GPU de los métodos es llevado a cabo y algunas directrices sobre su implementación son dados. Los esquemas numéricos antes mencionados se han aplicado a test de referencia académicos y a situaciones físicas más desafiantes como la simulación de tsunamis reales, y la comparación con datos de campo. Finalmente, un último capítulo es dedicado a medir la influencia al considerar efectos dispersivos en la simulación de transporte y arrastre de sedimentos. Para ello, se deduce un nuevo sistema de dos capas de aguas someras, se diseña un esquema numérico y se muestran algunos test académicos y de validación, que ofrecen resultados prometedores