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 new numerical scheme for a linear fluid–structure interaction problem

We consider a linear fluid–structure interaction problem consisting of the time-dependent Stokes equations coupled with those of linear elastodynamics. We assume that the fluid and the solid interact through a fixed interface. Then, we reformulate the problem following the ideas of [6], and propose a new monolithic method in terms of the velocity (both in the fluid and the solid) and the fluid pressure. We discretize the problem using the implicit Euler method for the time variable, piecewise linear elements in the solid and the mini-element in the fluid domain. Displacements in the structure can be recovered by means of a quadrature formula. Our numerical results confirm the robustness and good convergence properties of the proposed scheme. Moreover, our approach is easy to implement as compared with other methods available in the literature

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

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 $L^2$ 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

Investigation of infinite-dimensional dynamical system models applicable to granular flows

Recently Blackmore, Samulyak and Rosato developed a class of infinite-dimensional dynamical systems in the form of integro-partial differential equations, which have been called the BSR models. The BSR models were originally derived to model granular flows, but they actually have many additional applications in a variety of fields. BSR models have already been proven to be completely integrable infinite-dimensional Hamiltonian dynamical systems for perfectly elastic interactions in the case of one space dimension, but the well-posedness question of these systems is at least partially answered for the first time here. In particular, dynamical systems of the BSR type are proven to be well posed under mild auxiliary conditions and shown to have interesting properties. Also included is a novel derivation of a formula for (density) wave speeds in flow fields directly from the BSR model. In addition, an innovative semi-discrete numerical scheme for obtaining approximate solutions is described in detail and the questions of consistency, convergence, stability and accuracy of the scheme are treated at considerable length. It is shown how this numerical scheme can be used to help demonstrate the value of these models for predicting the evolution of granular flows and other flow field related phenomena, which is demonstrated to some extent by comparisons of the numerical results with experiments and some DEM simulations