753 research outputs found
A modified Galerkin/finite element method for the numerical solution of the Serre-Green-Naghdi system
A new modified Galerkin / Finite Element Method is proposed for the numerical
solution of the fully nonlinear shallow water wave equations. The new numerical
method allows the use of low-order Lagrange finite element spaces, despite the
fact that the system contains third order spatial partial derivatives for the
depth averaged velocity of the fluid. After studying the efficacy and the
conservation properties of the new numerical method, we proceed with the
validation of the new numerical model and boundary conditions by comparing the
numerical solutions with laboratory experiments and with available theoretical
asymptotic results
Boussinesq Systems of Bona-Smith Type on Plane Domains: Theory and Numerical Analysis
We consider a class of Boussinesq systems of Bona-Smith type in two space
dimensions approximating surface wave flows modelled by the three-dimensional
Euler equations. We show that various initial-boundary-value problems for these
systems, posed on a bounded plane domain are well posed locally in time. In the
case of reflective boundary conditions, the systems are discretized by a
modified Galerkin method which is proved to converge in at an optimal
rate. Numerical experiments are presented with the aim of simulating
two-dimensional surface waves in complex plane domains with a variety of
initial and boundary conditions, and comparing numerical solutions of
Bona-Smith systems with analogous solutions of the BBM-BBM system
Boussinesq-Peregrine water wave models and their numerical approximation
In this paper we consider the numerical solution of Boussinesq-Peregrine type
systems by the application of the Galerkin finite element method. The structure
of the Boussinesq systems is explained and certain alternative nonlinear and
dispersive terms are compared. A detailed study of the convergence properties
of the standard Galerkin method, using various finite element spaces on
unstructured triangular grids, is presented.
Along with the study of the Peregrine system, a new Boussinesq system of
BBM-BBM type is derived. The new system has the same structure in its momentum
equation but differs slightly in the mass conservation equation compared to the
Peregrine system. Further, the finite element method applied to the new system
has better convergence properties, when used for its numerical approximation.
Due to the lack of analytical formulas for solitary wave solutions for the
systems under consideration, a Galerkin finite element method combined with the
Petviashvili iteration is proposed for the numerical generation of accurate
approximations of line solitary waves. Various numerical experiments related to
the propagation of solitary and periodic waves over variable bottom topography
and their interaction with the boundaries of the domains are presented. We
conclude that both systems have similar accuracy when approximate long waves of
small amplitude while the Galerkin finite element method is more effective when
applied to BBM-BBM type systems
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
Fluid flow dynamics under location uncertainty
We present a derivation of a stochastic model of Navier Stokes equations that
relies on a decomposition of the velocity fields into a differentiable drift
component and a time uncorrelated uncertainty random term. This type of
decomposition is reminiscent in spirit to the classical Reynolds decomposition.
However, the random velocity fluctuations considered here are not
differentiable with respect to time, and they must be handled through
stochastic calculus. The dynamics associated with the differentiable drift
component is derived from a stochastic version of the Reynolds transport
theorem. It includes in its general form an uncertainty dependent "subgrid"
bulk formula that cannot be immediately related to the usual Boussinesq eddy
viscosity assumption constructed from thermal molecular agitation analogy. This
formulation, emerging from uncertainties on the fluid parcels location,
explains with another viewpoint some subgrid eddy diffusion models currently
used in computational fluid dynamics or in geophysical sciences and paves the
way for new large-scales flow modelling. We finally describe an applications of
our formalism to the derivation of stochastic versions of the Shallow water
equations or to the definition of reduced order dynamical systems
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