1,342 research outputs found
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
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
Finite volume schemes for dispersive wave propagation and runup
Finite volume schemes are commonly used to construct approximate solutions to
conservation laws. In this study we extend the framework of the finite volume
methods to dispersive water wave models, in particular to Boussinesq type
systems. We focus mainly on the application of the method to bidirectional
nonlinear, dispersive wave propagation in one space dimension. Special emphasis
is given to important nonlinear phenomena such as solitary waves interactions,
dispersive shock wave formation and the runup of breaking and non-breaking long
waves.Comment: 41 pafes, 20 figures. Other author's papers can be downloaded at
http://www.lama.univ-savoie.fr/~dutykh
Finite volume methods for unidirectional dispersive wave models
We extend the framework of the finite volume method to dispersive
unidirectional water wave propagation in one space dimension. In particular we
consider a KdV-BBM type equation. Explicit and IMEX Runge-Kutta type methods
are used for time discretizations. The fully discrete schemes are validated by
direct comparisons to analytic solutions. Invariants conservation properties
are also studied. Main applications include important nonlinear phenomena such
as dispersive shock wave formation, solitary waves and their various
interactions.Comment: 25 pages, 12 figures, 51 references. Other authors papers can be
downloaded at http://www.lama.univ-savoie.fr/~dutykh
Finite volume methods for unidirectional dispersive wave model
We extend the framework of the finite volume method to dispersive unidirectional water wave propagation in one space dimension. In particular, we consider a KdV–BBM-type equation. Explicit and implicit–explicit Runge–Kutta-type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariants’ conservation properties are also studied. Main applications include important nonlinear phenomena such as dispersive shock wave formation, solitary waves, and their various interaction
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, , and the layer distribution in water column becomes an optimizable parameter in this function. For a given tolerance dispersion error, the range of 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
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