176 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
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 and pseudo-spectral schemes for the fully nonlinear 1D Serre equations
After we derive the Serre system of equations of water wave theory from a
generalized variational principle, we present some of its structural
properties. We also propose a robust and accurate finite volume scheme to solve
these equations in one horizontal dimension. The numerical discretization is
validated by comparisons with analytical, experimental data or other numerical
solutions obtained by a highly accurate pseudo-spectral method.Comment: 28 pages, 16 figures, 75 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
Extended water wave systems of Boussinesq equations on a finite interval: Theory and numerical analysis
Considered here is a class of Boussinesq systems of Nwogu type. Such systems
describe propagation of nonlinear and dispersive water waves of significant
interest such as solitary and tsunami waves. The initial-boundary value problem
on a finite interval for this family of systems is studied both theoretically
and numerically. First, the linearization of a certain generalized Nwogu system
is solved analytically via the unified transform of Fokas. The corresponding
analysis reveals two types of admissible boundary conditions, thereby
suggesting appropriate boundary conditions for the nonlinear Nwogu system on a
finite interval. Then, well-posedness is established, both in the weak and in
the classical sense, for a regularized Nwogu system in the context of an
initial-boundary value problem that describes the dynamics of water waves in a
basin with wall-boundary conditions. In addition, a new modified Galerkin
method is suggested for the numerical discretization of this regularized system
in time, and its convergence is proved along with optimal error estimates.
Finally, numerical experiments illustrating the effect of the boundary
conditions on the reflection of solitary waves by a vertical wall are also
provided
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