Hyperbolic Conservation Laws

[no abstract available

A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes

In this paper, we introduce a discontinuous Finite Element formulation on simplicial unstructured meshes for the study of free surface flows based on the fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a new class of asymptotically equivalent equations, which have a simplified analytical structure, we consider a decoupling strategy: we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non-hydrostatic effects and we show that this source term can be computed through the resolution of scalar elliptic second-order sub-problems. The assets of the proposed discrete formulation are: (i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary order of approximation in space, (iii) the exact preservation of the motionless steady states, (iv) the preservation of the water height positivity, (v) a simple way to enhance any numerical code based on the nonlinear shallow water equations. The resulting numerical model is validated through several benchmarks involving nonlinear wave transformations and run-up over complex topographies

Institute for Computational Mechanics in Propulsion (ICOMP)

The Institute for Computational Mechanics in Propulsion (ICOMP) is operated by the Ohio Aerospace Institute (OAI) and funded under a cooperative agreement by the NASA Lewis Research Center in Cleveland, Ohio. The purpose of ICOMP is to develop techniques to improve problem-solving capabilities in all aspects of computational mechanics related to propulsion. This report describes the activities at ICOMP during 1994

An edge-based unstructured mesh discretisation in geospherical framework

An arbitrary finite-volume approach is developed for discretising partial differential equations governing fluid flows on the sphere. Unconventionally for unstructured-mesh global models, the governing equations are cast in the anholonomic geospherical framework established in computational meteorology. The resulting discretisation retains proven properties of the geospherical formulation, while it offers the flexibility of unstructured meshes in enabling irregular spatial resolution. The latter allows for a global enhancement of the spatial resolution away from the polar regions as well as for a local mesh refinement. A class of non-oscillatory forward-in-time edge-based solvers is developed and applied to numerical examples of three-dimensional hydrostatic flows, including shallow-water benchmarks, on a rotating sphere

Development and Optimization of Non-Hydrostatic Models for Water Waves and Fluid-Vegetation Interaction

The primary objective of this study is twofold: 1) to develop an efficient and accurate non-hydrostatic wave model for fully dispersive highly nonlinear waves, and 2) to investigate the interaction between waves and submerged flexible vegetation using a fully coupled wave-vegetation model. This research consists of three parts. Firstly, an analytical dispersion relationship is derived for waves simulated by models utilizing Keller-box scheme and central differencing for vertical discretization. The phase speed can be expressed as a rational polynomial function of the dimensionless water depth, $kh$, and the layer distribution in water column becomes an optimizable parameter in this function. For a given tolerance dispersion error, the range of $kh$ is extended and the layer thicknesses are optimally selected. The derived theoretical dispersion relationship is tested with linear and nonlinear standing waves generated by an Euler model. The optimization method is applicable to other non-hydrostatic models for water waves. Secondly, an efficient and accurate approach is developed to solve Euler equations for fully dispersive and highly nonlinear water waves. Discontinuous Galerkin, finite difference, and spectral element formulations are used for horizontal discretization, vertical discretization, and the Poisson equation, respectively. The Keller-box scheme is adopted for its capability of resolving frequency dispersion accurately with low vertical resolution (two or three layers). A three-stage optimal Strong Stability-Preserving Runge-Kutta (SSP-RK) scheme is employed for time integration. Thirdly, a fully coupled wave-vegetation model for simulating the interaction between water waves and submerged flexible plants is presented. The complete governing equation for vegetation motion is solved with a high-order finite element method and an implicit time differencing scheme. The vegetation model is fully coupled with a wave model to explore the relationship between displacement of water particle and plant stem, as well as the effect of vegetation flexibility on wave attenuation. This vegetation deformation model can be coupled with other wave models to simulate wave-vegetation interactions

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

Numerical modelling in a multiscale ocean

Systematic improvement in ocean modelling and prediction systems over the past several decades has resulted from several concurrent factors. The first of these has been a sustained increase in computational power, as summarized in Moore\u27s Law, without which much of this recent progress would not have been possible. Despite the limits imposed by existing computer hardware, however, significant accruals in system performance over the years have been achieved through novel innovations in system software, specifically the equations used to represent the temporal evolution of the oceanic state as well as the numerical solution procedures employed to solve them. Here, we review several recent approaches to system design that extend our capability to deal accurately with the multiple time and space scales characteristic of oceanic motion. The first two are methods designed to allow flexible and affordable enhancement in spatial resolution within targeted regions, relying on either a set of nested structured grids or, alternatively, a single unstructured grid. Finally, spatial discretization of the continuous equations necessarily omits finer, subgrid-scale processes whose effects on the resolved scales of motion cannot be neglected. We conclude with a discussion of the possibility of introducing subgrid-scale parameterizations to reflect the influences of unresolved processes