to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

https://oss.deltares.nl/documents/48999/59759/Roelvink_2009.pdf

Modelling storm impacts on beaches, dunes and barrier islands

SUMMARY
The open-source CFD library OpenFoam® contains a method for solving free surface Newtonian flows using the Reynolds averaged Navier–Stokes equations coupled with a volume of fluid method. In this paper, it is demonstrated how this has been extended with a generic wave generation and absorption method termed ‘wave relaxation zones’, on which a detailed account is given. The ability to use OpenFoam for the modelling of waves is demonstrated using two benchmark test cases, which show the ability to model wave propagation and wave breaking. Furthermore, the reflection coefficient from outlet relaxation zones is considered for a range of parameters. The toolbox is implemented in C++, and the flexibility in deriving new relaxation methods and implementing new wave theories along with other shapes of the relaxation zone is outlined. Subsequent to the publication of this paper, the toolbox has been made freely available through the OpenFoam-Extend Community. Copyright © 2011 John Wiley & Sons, Ltd.

A wave generation toolbox for the open‐source CFD library: OpenFoam®

In this paper we first describe the current method for obtaining the Camassa–Holm equation in the context of water waves; this requires a detour via the Green–Naghdi model equations, although the important connection with classical (Korteweg–de Vries) results is included. The assumptions underlying this derivation are described and their roles analysed. (The critical assumptions are, (i) the simplified structure through the depth of the water leading to the Green–Naghdi equations, and, (ii) the choice of submanifold in the Hamiltonian representation of the Green–Naghdi equations. The first of these turns out to be unimportant because the Green–Naghdi equations can be obtained directly from the full equations, if quantities averaged over the depth are considered. However, starting from the Green–Naghdi equations precludes, from the outset, any role for the variation of the flow properties with depth; we shall show that this variation is significant. The second assumption is inconsistent with the governing equations.)Returning to the full equations for the water-wave problem, we retain both parameters (amplitude, e, and shallowness, δ) and then seek a solution as an asymptotic expansion valid for, e → 0, δ → 0, independently. Retaining terms O(e), O(δ2) and O(eδ2), the resulting equation for the horizontal velocity component, evaluated at a specific depth, is a Camassa–Holm equation. Some properties of this equation, and how these relate to the surface wave, are described; the role of this special depth is discussed. The validity of the equation is also addressed; it is shown that the Camassa–Holm equation may not be uniformly valid: on suitably short length scales (measured by δ) other terms become important (resulting in a higher-order Korteweg–de Vries equation, for example). Finally, we indicate how our derivation can be extended to other scenarios; in particular, as an example, we produce a two-dimensional Camassa–Holm equation for water waves.

Camassa-Holm, Korteweg-de Vries and related models for water waves

A Boussinesq model for waves breaking in shallow water

Boussinesq–type equations of higher order in dispersion as well as in nonlinearity are derived for waves (and wave–current interaction) over an uneven bottom. Formulations are given in terms of various velocity variables such as the depth–averaged velocity and the particle velocity at the still water level, and at an arbitrary vertical location. The equations are enhanced and analysed with emphasis on linear dispersion, shoaling and nonlinear properties for large wave numbers.

As a starting point the velocity potential is expanded as a power series in the vertical coordinate measured from the still water level (SWL). Substituting this expansion into the Laplace equation leads to a velocity field expressed in terms of spatial derivatives of the vertical velocity ŵ and the horizontal velocity vector u at the SWL. The series expressions are given to infinite order in the dispersion parameter, μ. Satisfying the kinematic bottom boundary condition defines an implicit relation between ŵ and u, which is recast as an explicit recursive expression for ŵ in terms of u under the assumption that μ ≪ 1. Boussinesq equations are then derived from the dynamic and kinematic boundary conditions at the free surface. In this process the infinite series solutions are truncated at O (μ6), while all orders of the nonlinearity parameter, ϵ are included to that order in dispersion. This leads to a set of higher–order Boussinesq equations in terms of the surface elevation η and the horizontal velocity vector u at the SWL.

The equations are recast in terms of the depth–averaged velocity, U leaving out O (ϵ2μ4, which corresponds to assuming ϵ =O(μ). This formulation turns out to include singularities in linear dispersion as well as in nonlinearity. Next, the technique introduced by Madsen and others in 1991and Schaffer & Madsen in 1995 is invoked, and this results in aset of enhanced equations formulated in U and including O (μ4,ϵμ4)terms. These equations contain no singularities and the embedded linear and nonlinear properties are shown to be significantly improved. To quantify the accuracy, Stokes's third–order theory is used as reference and Fourier analyses of the new equations are carried out to third order (in nonlinearity) for regular waves on a constant depth and to first order for shoaling characteristics. Furthermore, analyses are carried out to second order for bichromatic waves and to first order for waves in ambient currents. These analyses are not restricted to small values of the linear dispersion parameter, μ. In conclusion, the new equations are shown to have linear dispersion characteristics corresponding to a Pade [4,4] expansion in k′h′ (wave number times depth) of the squared celerity according to Stokes's linear theory. This corresponds to a quite high accuracy in linear dispersion up to approximately k′h′ = 6. The high quality of dispersion is also achieved for the Doppler shift in connection with wave–current interaction and it allows for a study of wave blocking due to opposing currents. Also, the linear shoaling characteristics are shown to be excellent, and the accuracy of nonlinear transfer of energy to sub– and super–harmonics is found to be superior to previous formulations.

The equations are then recast in terms of the particle velocity, ũ, at an arbitrary vertical location including O μ4,ϵ5μ4)terms. This formulation includes, as special subsets, Boussinesq equations in terms of the bottom velocity or the surface velocity. Furthermore, the arbitrary location of the velocity variable can be used to optimize the embedded linear and nonlinear characteristics. A Fourier analysis is again carried out to third order (in ϵ) for regular waves. It turns out that Pade [4,4] linear dispersion characteristics can not be achieved for any choice of the location of the velocity variable. However, for an optimized location we achieve fairly good linear characteristics and very good nonlinear characteristics.

Finally, the formulation in terms of ũ is modified by introducing the technique of dispersion enhancement while retaining only O (μ4,ϵ5μ4) terms. Now the resulting set of equations do show Pade [4,4] dispersion characteristics in the case of pure waves as well as in connection with ambient currents, and again the nonlinear properties (such as second– and third–order transfer functions and amplitude dispersion) are shown to be superior to those of existing formulations of Boussinesq–type equations.

Higher–order Boussinesq–type equations for surface gravity waves: derivation and analysis

Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2008JC004932

Hemming A. Schäffer

Papers

A Boussinesq model for waves breaking in shallow water

Higher–order Boussinesq–type equations for surface gravity waves: derivation and analysis

Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves

On the solitary wave paradigm for tsunamis

Second-order wavemaker theory for irregular waves