MakisProf. Gerassimos Athanassoulis
School of Naval Architecture and Marine Engineering, National Technical University of Athens, Greece and
Research Center for High Performance Computing, ITMO University, St. Petersburg, Russia
A Hamiltonian approach to the nonlinear water-wave problem over bathymetry based on a new implementation of the DtN operator
Tuesday, May 10, 2016, 2:00pm to 3:00pm | Room 5-314

The fully nonlinear water-wave problem, over arbitrary smooth bathymetry, is formulated as a nonlinear, nonlocal Hamiltonian system, in the spirit of Zakharov [1] and Craig & Sulem [2] approach. The nonlocal coefficients of the Hamiltonian equations, which are usually expressed in terms of the Dirichlet-to-Neumann (DtN) operator [2], [3], are represented and implemented differently in this work. They are expressed by means of the first mode of a novel, rapidly convergent series expansion of the substrate wave potential, solved in the instantaneous geometrical configuration [4], [5]. This expansion is based on a local vertical basis consisting of the usual vertical modes (providing an L2-local vertical basis), plus two additional modes, which make it an H2-local (Sobolev) vertical basis at each horizontal position [5], [6], [7]. Thus, a highly accurate representation/implementation of the boundary derivatives of the wave potential is possible, even for very steep waves and bathymetries. Besides, the expansion is rapidly convergent, with exponential decay rate for the few first modes, and O(n-4) decay rate asymptotically. As a consequence, only a few modes, up to 6 or 7, are enough to provide a highly accurate representation of the nonlocal coefficients of the Hamiltonian equations. The latter are solved in the time domain by the classical, fourth-order Runge-Kutta method, providing nice solutions to various configurations. The excitation to simulations of wave propagation over intermediate-depth water waves is applied to a generation-absorption entrance layer [8].

Numerical results for the standard Beji-Battjes experiment show excellent agreement with measurements at all stations [4]. Further, numerical results are shown for nonlinear Bragg scattering, solitary wave propagation and collision over flat bottom, reflection of a solitary wave by a vertical wall, as well as solitary wave propagation over a sloped bottom, an abrupt step, and an undulating bottom. Comparisons with experiments and/or other solution methods are presented whenever possible. Initial conditions for the solitary-wave related numerical experiments are calculated by using "exact" numerical solutions to the solitary wave problem in constant depth [9].

[1] V. Zakharov, "Stability of periodic waves of finite amplitude on the surface of a deep fluid," Zhurnal Prikl. Mekhaniki i Tekhnicheskoi Fiz., vol. 9, no. 2, pp. 86–94, 1968.
[2] W. Craig and C. Sulem, "Numerical Simulation of Gravity Waves," J. Comput. Phys., vol. 108, pp. 73–83, 1993.
[3] L. Xu and P. Guyenne, "Numerical simulation of three-dimensional nonlinear water waves," J. Comput. Phys., vol. 228, no. 22, pp. 8446–8466, Dec. 2009.
[4] G. A. Athanassoulis and C. E. Papoutsellis, "New form of the Hamiltonian equations for the nonlinear water-wave problem, based on a new representation of the DtN operator, and some applications," in Proceedings of the 34th International Conference on Ocean, Offshore and Arctic Engineering, 2015.
[5] G. A. Athanassoulis and C. E. Papoutsellis, "Rapidly convergent series expansions in waveguides with nonplanar boundaries, with an application to the computation of Dirichlet to Neumann operator," 2016. Submitted for publication
[6] G. Athanassoulis and K. Belibassakis, "A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions," J. Fluid Mech., vol. 389, pp. 275–301, 1999.
[7] G. Athanassoulis and K. Belibassakis, "Rapidly-Convergent Local-Mode Representations for Wave Propagation and Scattering Curved-Boundary Waveguides," in 6th International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES 2003), 2003, pp. 451–456.
[8] Y. Zhang, A. B. Kennedy, N. Panda, C. Dawson, and J. J. Westerink, "Generating-absorbing sponge layers for phase-resolving wave models," Coast. Eng., vol. 84, pp. 1–9, 2014.
[9] D. Clamond and D. Dutykh, "Fast accurate computation of the fully nonlinear solitary surface gravity waves," Comput. Fluids, vol. 84, pp. 35–38, Sep. 2013.

Short Bio:

Dr. Gerassimos Athanassoulis is Professor in the School of Naval Architecture and Marine Engineering, Section of Ship and Marine Hydrodynamics, at the National Technical University of Athens. He is also research Professor at ITMO University, St. Petersburg, Russia. He holds a Diploma in Naval Architecture and Marine Engineering from the National Technical University of Athens (1977) and a PhD from the same University (1982). His research interests include: Ship and Marine Hydrodynamics, Wave Phenomena in the Sea, Variational Principles for Mechanical and Coupled Systems, and Stochastic modelling of Environmental Parameters and Dynamical Systems. He is the author of about 60 Journal Publications and more than 120 Conference Presentations, mainly referring to the aforementioned subjects.