16/07/2015, 15:00 — 16:00 — Room P3.10, Mathematics Building
Nuno Lopes, Instituto Superior de Engenharia de Lisboa
Analytical and Numerical Methods of the type FEM-C/D for Improved Boussinesq Models
In this talk, some analytical and numerical models are developed for the generation and propagation of surface water waves. These problems are solved using asymptotic and numerical methods. Regarding the numerical methods, we consider the continuous and continuous/discontinuous Galerkin finite element methods (FEM-C/D) with penalty terms. In the first problem, the model of Zhao et al. (2004) is extended in order to include some effects like dissipation and absorption of the energy of the surface water waves. We show that this model is robust with respect to the instabilities related to steep bottom gradients of the bathymetry. A new class of nonlinear Boussinesq-type systems is derived in the second problem. A CFL type condition is obtained for the linearized problem with constant bathymetry. The consistency of the dispersion relation as well as the good stability properties of this model are verified. From the numerical tests, we can conclude that the proposed numerical model is appropriate to model surface water waves. In the third problem, a class of Korteweg, de Vries–Benjamin, Bona and Mahony (KdV-BBM) type equations is deduced. The Nwogu’s parameter is determined in order to optimise the velocity potential of the linearized KdV-BBM model. Moreover, a numerical analysis of the proposed model is performed. We conclude that the KdV-BBM model is less prone to instabilities than the KdV model. Finally, a new Boussinesq-type differential equation of sixth-order to model bidirectional waves is derived and exact travelling wave solutions are obtained. A new analytical travelling wave solution is found. This is a joint work with P. J. S. Pereira and L. Trabucho.
