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

Abstract

Abstract. 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. Introduction. Accurate modeling of surface water wave dynamics over bottom topography is of great importance to coastal engineers and has drawn considerable attention in recent years. Here are just a few applications: linear [16,33] and nonlinear wave shoaling [25,24,28,27], Bragg reflection [35,15,36,14,30,33,4], harmonic wave generation [3,17,18], and tsunami generation [37,22]. As the publications cited above attest, the character of coastal wave dynamics can be very complex: when entering shallow water, waves are strongly affected by the bottom through shoaling, refraction, diffraction, and reflection. Nonlinear effects related to wave-wave and wave-bottom interactions can cause wave scattering and depth-induced wave breaking. In turn, the resulting nonlinear waves can have a great influence on sediment transport and the formation of sandbars in nearshore regions. The presence of bottom topography also introduces additional space and time scales to the classical perturbation problem (see, e.g., [10]).Traditionally, the water wave problem on variable depth has been modeled using long-wave approximations such as the Boussinesq or shallow water equations [26,40]. See also [39] for a weakly nonlinear formulation for water waves over topography. In recent years, progress in both mathematical techniques and computer power has led to a rapid development of numerical models solving the full Euler equations. These can be divided into two main categories: boundary integral methods [2,24,49,7,20,27] and spectral methods. In particular, efficient spectral methods based on perturbation expansions have been developed for the computation of water waves on constant or

Numerical simulation of three-dimensional nonlinear water waves

“…Possible extensions of the present model include three-dimensional surface waves over a variable bottom topography, for which a modification of this method is suitable [15,32], and the development of an optimized/parallelized code [46] in order to make large-scale problems (with a large number of grid points) more tractable. These directions of inquiry are envisioned for future work.…”

“…This formulation of the water wave problem is convenient for the numerical simulation of three-dimensional nonlinear waves as presented in this paper, as well as in a number of other settings. These include numerical and asymptotic studies of surface waves over a rough bottom [15,32,7], and waves at the interface between immiscible fluids [14]. It is also understood that overturning waves (i.e.…”

“…Each term in this Taylor series can be obtained from a recursion formula and can be efficiently computed by a pseudospectral method using the fast Fourier transform. Here we will only pay attention to surface water waves propagating in deep water or on uniform depth, although this is not a restriction of the model which can be extended to include a variable bottom topography [15,32]. The latter references consider smooth topographies of small amplitude.…”

Stable computation of the functional variation of the Dirichlet–Neumann operator

“…[27]. With (12)- (15) in hand the TFE method now instructs us to make the following Taylor series expansions for the field and DNO uðx; y; eÞ ¼…”

“…Methods based upon integral equations, finite differences, and finite elements are all available, however, if the shape of the geometry is a small deformation of a simple one (e.g., a separable domain such as a rectangle in two dimensions) then an algorithm based upon perturbation series is natural. For the computation of DNO, several Boundary Perturbation algorithms are available including the methods of Operator Expansions (OE) [1,[10][11][12][13][14][15], Field Expansions (FE) [16][17][18][19][20][21][22], and Transformed Field Expansions (TFE) [23][24][25]. The former two methods are easy to implement, highly accurate within their domain of applicability (e.g., the excellent agreement between the numerical simulations of [13,15] and wave tank experiments), and have computational complexity competitive with state-of-the-art integral equation solvers [26].…”