482 research outputs found
A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: Triangular grids
A novel wetting and drying treatment for second-order Runge-Kutta
discontinuous Galerkin (RKDG2) methods solving the non-linear shallow water
equations is proposed. It is developed for general conforming two-dimensional
triangular meshes and utilizes a slope limiting strategy to accurately model
inundation. The method features a non-destructive limiter, which concurrently
meets the requirements for linear stability and wetting and drying. It further
combines existing approaches for positivity preservation and well-balancing
with an innovative velocity-based limiting of the momentum. This limiting
controls spurious velocities in the vicinity of the wet/dry interface. It leads
to a computationally stable and robust scheme -- even on unstructured grids --
and allows for large time steps in combination with explicit time integrators.
The scheme comprises only one free parameter, to which it is not sensitive in
terms of stability. A number of numerical test cases, ranging from analytical
tests to near-realistic laboratory benchmarks, demonstrate the performance of
the method for inundation applications. In particular, super-linear
convergence, mass-conservation, well-balancedness, and stability are verified
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
A Derivative Recovery Spectral Volume model for the analysis of constituents transport in one-dimensional flows
The treatment of advective fluxes in high-order finite volume models is well established, but this is not the case for diffusive fluxes, due to the conflict between the discontinuous representation of the solution and the continuous structure of analytic solutions. In this paper, a derivative reconstruction approach is proposed in the context of spectral volume methods, for the approximation of diffusive fluxes, aiming at the reconciliation of this conflict. Two different reconstructions are used for advective and diffusive fluxes: the advective reconstruction makes use of the information contained in a spectral cell, and allows the formation of discontinuities at the spectral cells boundaries; the diffusive reconstruction makes use of the information contained in contiguous spectral cells, imposing the continuity of the reconstruction at the spectral cells boundaries. The method is demonstrated by a number of numerical experiments, including the solution of shallow-water equations, complemented with the advective-diffusive transport equation of a conservative substance, showing the promising abilities of the numerical scheme proposed
A large time-step and well-balanced Lagrange-Projection type scheme for the shallow-water equations
This work focuses on the numerical approximation of the Shallow Water
Equations (SWE) using a Lagrange-Projection type approach. We propose to extend
to this context recent implicit-explicit schemes developed in the framework of
compressibleflows, with or without stiff source terms. These methods enable the
use of time steps that are no longer constrained by the sound velocity thanks
to an implicit treatment of the acoustic waves, and maintain accuracy in the
subsonic regime thanks to an explicit treatment of the material waves. In the
present setting, a particular attention will be also given to the
discretization of the non-conservative terms in SWE and more specifically to
the well-known well-balanced property. We prove that the proposed numerical
strategy enjoys important non linear stability properties and we illustrate its
behaviour past several relevant test cases
A "well-balanced" finite volume scheme for blood flow simulation
We are interested in simulating blood flow in arteries with a one dimensional
model. Thanks to recent developments in the analysis of hyperbolic system of
conservation laws (in the Saint-Venant/ shallow water equations context) we
will perform a simple finite volume scheme. We focus on conservation properties
of this scheme which were not previously considered. To emphasize the necessity
of this scheme, we present how a too simple numerical scheme may induce
spurious flows when the basic static shape of the radius changes. On contrary,
the proposed scheme is "well-balanced": it preserves equilibria of Q = 0. Then
examples of analytical or linearized solutions with and without viscous damping
are presented to validate the calculations. The influence of abrupt change of
basic radius is emphasized in the case of an aneurism.Comment: 36 page
Well-balanced fifth-order finite difference Hermite WENO scheme for the shallow water equations
In this paper, we propose a well-balanced fifth-order finite difference
Hermite WENO (HWENO) scheme for the shallow water equations with non-flat
bottom topography in pre-balanced form. For achieving the well-balance
property, we adopt the similar idea of WENO-XS scheme [Xing and Shu, J. Comput.
Phys., 208 (2005), 206-227.] to balance the flux gradients and the source
terms. The fluxes in the original equation are reconstructed by the nonlinear
HWENO reconstructions while other fluxes in the derivative equations are
approximated by the high-degree polynomials directly. And an HWENO limiter is
applied for the derivatives of equilibrium variables in time discretization
step to control spurious oscillations which maintains the well-balance
property. Instead of using a five-point stencil in the same fifth-order WENO-XS
scheme, the proposed HWENO scheme only needs a compact three-point stencil in
the reconstruction. Various benchmark examples in one and two dimensions are
presented to show the HWENO scheme is fifth-order accuracy, preserves
steady-state solution, has better resolution, is more accurate and efficient,
and is essentially non-oscillatory.Comment: 24 pages, 11 figure
- …