AMCS Mini Course: Mathematical and numerical modelling of water waves

Overview

Abstract : The mathematical modelling of water waves continues to attract great interest both from a mathematical point of view, due to the difficulty of solving the systems of nonlinear partial differential equations that describe the relevant physical phenomena, and also from that of fluid mechanics, due to complexity of the physical phenomena that occur during the propagation of water waves and their importance in oceanography and coastal dynamics. The basic mathematical model is the system of the Euler equation for the motion of the free surface and the evolution of the velocity field of the water. The seminar consists of three parts and selected topics related to the mathematical modelling of water waves will be presented. In particular, in the first part of the seminar we will present the derivation of the Euler equations from basic principles of fluid mechanics. In order to overcome certain difficulties in handling the Euler equations we will proceed with certain simplifications based on asymptotic reasoning to derive simplified model equations for long water waves of small amplitude. The simplified systems are known to as Boussinesq systems and some of them appeared to have favourable properties for the approximation of long waves such as tsunamis and solitary waves. Approximate asymptotic equations derived from the Euler equations are usually heuristic, in the sense that although the derivation is formal they might not preserve basic mathematical or physical properties of the original equations. For this reason the second part of the seminar focuses on the mathematical theory of certain Boussinesq systems. We will present the well-posedness theory for the Cauchy (initial-value) problem and we will reject the ill-posed systems as questions on their accuracy will be risen. Finally we will discuss the existence of classical solitary wave solutions for the various Boussinesq models since they must retain this fundamental property of the Euler equations. Based on these theoretical developments we will categorise the Boussinesq systems as good or bad models for long surface water waves of small amplitude. In the last part of the seminar we will present the numerical solution of some Boussinesq systems using the standard Galerkin/Finite Element Method. Some numerical experiments related to the properties of the solitary waves will be presented.

Bio: Dimitrios Mitsotakis is a Mathematician, graduated from the University of Crete with the highest honors in the year 2000. He received a master’s degree in Applied and Numerical Analysis in the year 2003 and a PhD in Mathematics in 2007 from the University of Athens. He is skilled in high performance computing at the Edinburgh Parallel Computing Center at The University of Edinburgh in the year 2000. Dimitrios visited the Université Paris-Sud (2008-2010) as a Marie Curie researcher, the IMA Postdoctoral Associate, at the University of Minnesota (2010-2012) as an associate postdoc and the University of California, Merced (2012-2014) as a visiting Assistant Professor. Dimitrios is currently a Senior Lecturer at the School of Mathematics and Statistics of Victoria University of Wellington. Dimitrios is interested in the theory of water waves and in the numerical analysis of Partial Differential Equations. He develops numerical methods for the solution of models for water waves and he studies real-world applications such as the generation of tsunamis. Some of his main contributions are in the theory of Boussinesq systems for nonlinear and dispersive water waves. He has also developed numerical models for the simulation of the generation and propagation of tsunami waves and the dispersive run-up. He is also interested in interfacial water waves and waves in superfluids. Numerical methods of preference are Finite Element, Spectral, Finite Volume and Discontinuous Galerkin methods.