A High-Order Spectral Method for Nonlinear Water Waves over Moving Bottom Topography

We present a numerical method for simulations of nonlinear surface water waves over variable bathymetry. It is applicable to either two- or three-dimensional flows, as well as to either static or moving bottom topography. The method is based on the reduction of the problem to a lower-dimensional Hamiltonian system involving boundary quantities alone. A key component of this formulation is the Dirichlet-Neumann operator which, in light of its joint analyticity properties with respect to surface and bottom deformations, is computed using its Taylor series representation. We present new, stabilized forms for the Taylor terms, each of which is efficiently computed by a pseudospectral method using the fast Fourier transform. Physically relevant applications are displayed to illustrate the performance of the method; comparisons with analytical solutions and laboratory experiments are provided.

[1]  H. Bingham,et al.  A Fourier–Boussinesq method for nonlinear water waves , 2005 .

[2]  Walter Craig,et al.  Numerical simulation of gravity waves , 1993 .

[3]  David R. Basco,et al.  Water Wave Mechanics for Engineers and Scientists , 1985 .

[4]  John Grue,et al.  An efficient model for three-dimensional surface wave simulations , 2005 .

[5]  Ralph A. Smith,et al.  An operator expansion formalism for nonlinear surface waves over variable depth , 1998, Journal of Fluid Mechanics.

[6]  Denys Dutykh,et al.  Water waves generated by a moving bottom , 2007 .

[7]  M. W. Dingemans,et al.  Water Wave Propagation Over Uneven Bottoms , 1997 .

[8]  Dick K. P. Yue,et al.  Computations of fully nonlinear three-dimensional wave–wave and wave–body interactions. Part 2. Nonlinear waves and forces on a body , 2001, Journal of Fluid Mechanics.

[9]  David Lannes,et al.  Well-posedness of the water-waves equations , 2005 .

[10]  H. J.,et al.  Hydrodynamics , 1924, Nature.

[11]  Philippe Guyenne,et al.  Hamiltonian long‐wave expansions for free surfaces and interfaces , 2005 .

[12]  Philippe Guyenne,et al.  Hamiltonian long–wave expansions for water waves over a rough bottom , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[13]  Chiang C. Mei,et al.  Resonant reflection of surface water waves by periodic sandbars , 1985, Journal of Fluid Mechanics.

[14]  C. Mei.  The applied dynamics of ocean surface waves , 1983 .

[15]  D. Milder,et al.  AN IMPROVED FORMALISM FOR WAVE SCATTERING FROM ROUGH SURFACES , 1991 .

[16]  J. Kirby.  A note on Bragg scattering of surface waves by sinusoidal bars , 1993 .

[17]  C. Swan,et al.  On the efficient numerical simulation of directionally spread surface water waves , 2001 .

[18]  J. Thomas Beale,et al.  A convergent boundary integral method for three-dimensional water waves , 2001, Math. Comput..

[19]  Dick K. P. Yue,et al.  On generalized Bragg scattering of surface waves by bottom ripples , 1998, Journal of Fluid Mechanics.

[20]  Roger H.J. Grimshaw,et al.  The solitary wave in water of variable depth , 1970, Journal of Fluid Mechanics.

[21]  John Grue,et al.  An efficient model for three-dimensional surface wave simulations. Part II: Generation and absorption , 2005 .

[22]  David P. Nicholls,et al.  Traveling Water Waves , 1998 .

[23]  Robert A. Dalrymple,et al.  Water Waves Over Ripples , 1986 .

[24]  Alan G. Davies,et al.  Surface-wave propagation over sinusoidally varying topography , 1984, Journal of Fluid Mechanics.

[25]  Stephan T. Grilli,et al.  Numerical study of three-dimensional overturning waves in shallow water , 2006, Journal of Fluid Mechanics.

[26]  A. Calderón,et al.  Cauchy integrals on Lipschitz curves and related operators. , 1977, Proceedings of the National Academy of Sciences of the United States of America.

[27]  I. A. Svendsen,et al.  BREAKING CRITERION AND CHARACTERISTICS FOR SOLITARY WAVES ON SLOPES , 1997 .

[28]  Philippe Guyenne,et al.  Numerical simulation of solitary waves on plane slopes , 2005, Math. Comput. Simul..

[29]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[30]  Walter Craig,et al.  Traveling Two and Three Dimensional Capillary Gravity Water Waves , 2000, SIAM J. Math. Anal..

[31]  D. Gottlieb,et al.  Numerical analysis of spectral methods , 1977 .

[32]  Stephan T. Grilli,et al.  A fully non‐linear model for three‐dimensional overturning waves over an arbitrary bottom , 2001 .

[33]  Jurjen A. Battjes,et al.  Experimental investigation of wave propagation over a bar , 1993 .

[34]  Steven G. Johnson,et al.  The Fastest Fourier Transform in the West , 1997 .

[35]  Dick K. P. Yue,et al.  A high-order spectral method for the study of nonlinear gravity waves , 1987, Journal of Fluid Mechanics.

[36]  Fernando Reitich,et al.  A new approach to analyticity of Dirichlet-Neumann operators , 2001, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[37]  André Nachbin,et al.  A Terrain-Following Boussinesq System , 2003, SIAM J. Appl. Math..

[38]  M. W. Dingemans,et al.  Comparison of computations with Boussinesq-like models and laboratory measurements , 1994 .

[39]  Walter Craig,et al.  The modulational regime of three-dimensional water waves and the Davey-Stewartson system , 1997 .

[40]  Per A. Madsen,et al.  A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry , 2006 .

[41]  Vladimir E. Zakharov,et al.  Stability of periodic waves of finite amplitude on the surface of a deep fluid , 1968 .

[42]  Bei Hu,et al.  Analyticity of Dirichlet-Neumann Operators on Hölder and Lipschitz Domains , 2005, SIAM J. Math. Anal..

[43]  C. Canuto.  Spectral methods in fluid dynamics , 1991 .

[44]  Ronald R. Coifman,et al.  Nonlinear harmonic analysis and analytic dependence , 1985 .

[45]  Bruce J. West,et al.  A new numerical method for surface hydrodynamics , 1987 .

[46]  D. Nicholls,et al.  Joint Analyticity and Analytic Continuation of Dirichlet–Neumann Operators on Doubly Perturbed Domains , 2008 .

[47]  Fernando Reitich,et al.  Digital Object Identifier (DOI) 10.1007/s002110200399 Analytic continuation of Dirichlet-Neumann operators , 2022 .

[48]  P. Milewski.  A Formulation for Water Waves over Topography , 1998 .

[49]  Frédéric Dias,et al.  A fast method for nonlinear three-dimensional free-surface waves , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[50]  E. Mansard,et al.  The Measurement of Incident and Reflected Spectra Using a Least squares Method , 1980 .

[51]  Fernando Reitich,et al.  Stability of High-Order Perturbative Methods for the Computation of Dirichlet-Neumann Operators , 2001 .