Numerical analysis of time-dependent Boussinesq models

In this paper we analyse numerical models for time-dependent Boussinesq equations. These equations arise when so-called Boussinesq terms are introduced into the shallow water equations. We use the Boussinesq terms proposed by Katapodes and Dingemans. These terms generalize the constant depth terms given by Broer. The shallow water equations are discretized by using fourth-order finite difference formulae for the space derivatives and a fourth-order explicit time integrator. The effect on the stability and accuracy of various discrete Boussinesq terms is investigated. Numerical experiments are presented in the case of a fourth-order Runge-Kutta time integrator.     

Instability of solitary waves for generalized Boussinesq equations

An equation of Boussinesq-type of the formu _tt -u _xx +(f(u)+u_xx)_xx=0 is considered. It is shown that a traveling wave may be stable or unstable, depending on the range of the wave's speed of propagation and on the nonlinearity. Sharp conditions to that effect are given.This research is supported in part by NSF Grant DMS 90-23864.     

On uniformly accurate high‐order Boussinesq difference equations for water waves

A new accurate finite‐difference (AFD) numerical method is developed specifically for solving high‐order Boussinesq (HOB) equations. The method solves the water‐wave flow with much higher accuracy compared to the standard finite‐difference (SFD) method for the same computer resources. It is first developed for linear water waves and then for the nonlinear problem. It is presented for a horizontal bottom, but can be used for variable depth as well. The method can be developed for other equations as long as they use Padé approximation, for example extensions of the parabolic equation for acoustic wave problems. Finally, the results of the new method and the SFD method are compared with the accurate solution for nonlinear progressive waves over a horizontal bottom that is found using the stream function theory. The agreement of the AFD to the accurate solution is found to be excellent compared to the SFD solution. Copyright © 2005 John Wiley & Sons, Ltd.     

An Accurate Finite Difference Scheme for Boussinesq Equations

In this paper, a stable and accurate finite difference scheme using a space-staggered grid is proposed for solving the extended Boussinesq-type equations as derived by Nwogu [Journal of Waterway, Port Coastal and Ocean Engineering, ASCE, 119, (1993) 618–638]. The alternate direction iterative method combined with an efficient predictor–corrector scheme is adopted for the numerical solution of the governing differential equations. The proposed method is verified by two test cases where experimental data are available for comparison. The first case is wave focusing by bottom topography as studied by Whalin [The limit of applicability of linear wave refraction theory in a convergence zone. Res. Rep. H-71-3, U.S.Army Corps of Engrs. Waterways Expt. Station, Vicksburg (1971)]. The second case is wave runup around a circular cylinder as investigated experimentally by Isaacson (Journal of the Waterway, Port, Coastal and Ocean Division, ASCE, 104, (1978), 69–79). Numerical results agree very well with the corresponding experimental data in both cases.     

Boussinesq cut‐cell model for non‐linear wave interaction with coastal structures

Boussinesq models describe the phase‐resolved hydrodynamics of unbroken waves and wave‐induced currents in shallow coastal waters. Many enhanced versions of the Boussinesq equations are available in the literature, aiming to improve the representation of linear dispersion and non‐linearity. This paper describes the numerical solution of the extended Boussinesq equations derived by Madsen and Sørensen (Coastal Eng. 1992; 15 :371–388) on Cartesian cut‐cell grids, the aim being to model non‐linear wave interaction with coastal structures. An explicit second‐order MUSCL‐Hancock Godunov‐type finite volume scheme is used to solve the non‐linear and weakly dispersive Boussinesq‐type equations. Interface fluxes are evaluated using an HLLC approximate Riemann solver. A ghost‐cell immersed boundary method is used to update flow information in the smallest cut cells and overcome the time step restriction that would otherwise apply. The model is validated for solitary wave reflection from a vertical wall, diffraction of a solitary wave by a truncated barrier, and solitary wave scattering and diffraction from a vertical circular cylinder. In all cases, the model gives satisfactory predictions in comparison with the published analytical solutions and experimental measurements. Copyright © 2007 John Wiley & Sons, Ltd.     

A finite element method for the one‐dimensional extended Boussinesq equations

A new finite element method for Nwogu's (O. Nwogu, ASCE J. Waterw., Port, Coast., Ocean Eng., 119 , 618–638 (1993)) one‐dimensional extended Boussinesq equations is presented using a linear element spatial discretisation method coupled with a sophisticated adaptive time integration package. The accuracy of the scheme is compared to that of an existing finite difference method (G. Wei and J.T. Kirby, ASCE J. Waterw., Port, Coast., Ocean Eng., 121 , 251–261 (1995)) by considering the truncation error at a node. Numerical tests with solitary and regular waves propagating in variable depth environments are compared with theoretical and experimental data. The accuracy of the results confirms the analytical prediction and shows that the new approach competes well with existing finite difference methods. The finite element formulation is shown to enable the method to be extended to irregular meshes in one dimension and has the potential to allow for extension to the important practical case of unstructured triangular meshes in two dimensions. This latter case is discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

Surface waves propagation in shallow water: A finite element model

A two-dimensional (in-plane) numerical model for surface waves propagation based on the non-linear dispersive wave approach described by Boussinesq-type equations, which provide an attractive theory for predicting the depth-averaged velocity field resulting from that wave-type propagation in shallow water, is presented. The numerical solution of the corresponding partial differential equations by finite-difference methods has been the subject of several scientific works. In the present work we propose a new approach to the problem: the spatial discretization of the system composed by the Boussinesq equations is made by a finite element method, making use of the weighted residual technique for the solution approach within each element. The model is validated by comparing numerical results with theoretical solutions and with results obtained experimentally.     

A finite element method for the two‐dimensional extended Boussinesq equations

A new numerical method for Nwogu's (ASCE Journal of Waterway, Port, Coastal and Ocean Engineering 1993; 119 :618)two‐dimensional extended Boussinesq equations is presented using a linear triangular finite element spatial discretization coupled with a sophisticated adaptive time integration package. The authors have previously presented a finite element method for the one‐dimensional form of these equations (M. Walkley and M. Berzins (International Journal for Numerical Methods in Fluids 1999; 29 (2):143)) and this paper describes the extension of these ideas to the two‐dimensional equations and the application of the method to complex geometries using unstructured triangular grids. Computational results are presented for two standard test problems and a realistic harbour model. Copyright © 2002 John Wiley & Sons, Ltd.     

Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations

We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scale expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg-de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes.     

Exact expressions for transient forced internal gravity waves and spatially-localized wave packets in a Boussinesq fluid

We re-examine a simple model describing the propagation of transient forced internal gravity waves in a Boussinesq fluid with constant horizontal mean velocity which was previously studied by Nadon and Campbell (Wave Motion, 2007). The waves are generated by a horizontally-periodic lower boundary condition and propagate upwards. We derive an alternative exact expression for the solution which more readily gives insight into the behaviour of the solution at high altitude. Some special cases of lower boundary conditions are considered to illustrate the features of the solution. This form of the solution allows us to use a Fourier transform to derive the solution for the more general situation where a wave packet is generated by a horizontally-localized lower boundary condition, comprising a continuous spectrum of horizontal wavenumbers or Fourier modes. This is a more realistic representation of internal gravity waves in the atmosphere and can be used as a starting point for investigating waves generated by an obstacle of finite horizontal extent such as an isolated mountain or a mountain range.     

High‐order compact scheme for Boussinesq equations: implementation and numerical boundary condition issue

Application of the three‐point fourth‐order compact scheme to spatial differencing of the vorticity‐stream function‐density formulation of the two‐dimensional incompressible Boussinesq equations is presented. The details for the derivation of difference relations at boundaries to generate accurate and stable solutions are also given. To assess the numerical accuracy, two linear prototype test problems with known exact solution are used. The two‐dimensional planar and cylindrical lock‐exchange flow configurations are used to conduct the numerical experiments for the Boussinesq equations. Quantitative measures for the two linear prototype test problems and comparison of the results of this work with the published results for the planar lock‐exchange flow indicates the validity and accuracy of the three‐point fourth‐order compact scheme for numerical solution of two‐dimensional incompressible Boussinesq equations. In addition, the study of using different high‐order numerical boundary conditions for the implementation of the no‐penetration boundary condition for the density at no‐slip walls is considered. It is shown that the numerical solution is sensitive to the choice of difference relation for the density at boundaries and using an inappropriate difference relation leads to spurious numerical solution. Copyright © 2011 John Wiley & Sons, Ltd.     

Bidirectional Whitham equations as models of waves on shallow water

Hammack & Segur (1978) conducted a series of surface water-wave experiments in which the evolution of long waves of depression was measured and studied. This present work compares time series from these experiments with predictions from numerical simulations of the KdV, Serre, and five unidirectional and bidirectional Whitham-type equations. These comparisons show that the most accurate predictions come from models that contain accurate reproductions of the Euler phase velocity, sufficient nonlinearity, and surface tension effects. The main goal of this paper is to determine how accurately the bidirectional Whitham equations can model data from real-world experiments of waves on shallow water. Most interestingly, the unidirectional Whitham equation including surface tension provides the most accurate predictions for these experiments. If the initial horizontal velocities are assumed to be zero (the velocities were not measured in the experiments), the three bidirectional Whitham systems examined herein provide approximations that are significantly more accurate than the KdV and Serre equations. However, they are not as accurate as predictions obtained from the unidirectional Whitham equation.     

Higher order Boussinesq-type equations for water waves on uneven bottom

Higher order Boussinesq-type equations for wave propagation over variable bathymetry were derived. The time dependent free surface boundary conditions were used to compute the change of the free surface in time domain. The free surface velocities and the bottom velocities were connected by the exact solution of the Laplace equation. Taking the velocities on half relative water depth as the fundamental unknowns, terms relating to the gradient of the water depth were retained in the inverse series expansion of the exact solution, with which the problem was closed. With enhancements of the finite order Taylor expansion for the velocity field, the application range of the present model was extended to the slope bottom which is not so mild. For linear properties, some validation computations of linear shoaling and Booij' s tests were carried out. The problems of wave-current interactions were also studied numerically to test the performance of the enhanced Boussinesq equations associated with the effect of currents. All these computational results confirm perfectly to the theoretical solution as well as other numerical solutions of the full potential problem available.     

A parametric finite‐difference method for shallow sea waves

This paper presents a parametric finite‐difference scheme concerning the numerical solution of the one‐dimensional Boussinesq‐type set of equations, as they were introduced by Peregrine (J. Fluid Mech. 1967; 27 (4)) in the case of waves relatively long with small amplitudes in water of varying depth. The proposed method, which can be considered as a generalization of the Crank‐Nickolson method, aims to investigate alternative approaches in order to improve the accuracy of analogous methods known from bibliography. The resulting linear finite‐difference scheme, which is analysed for stability using the Fourier method, has been applied successfully to a problem used by Beji and Battjes (Coastal Eng. 1994; 23 : 1–16), giving numerical results which are in good agreement with the corresponding results given by MIKE 21 BW (User Guide. In: MIKE 21, Wave Modelling, User Guide. 2002; 271–392) developed by DHI Software. Copyright © 2006 John Wiley & Sons, Ltd.     

On the shoreline boundary conditions for Boussinesq‐type models

We propose and illustrate a novel type of shoreline boundary conditions for Boussinesq‐type models. On the basis of characteristic equations of the non‐linear shallow water equations, boundary conditions are developed equations that can suitably model the motion of the instantaneous shoreline. Such boundary conditions are then implemented in a numerical solver for a specific set of Boussinesq‐type equations, which have been proved very effective for near‐shore modelling. Finally, a number of tests are performed to validate and illustrate the behaviour of the new conditions. Copyright © 2001 John Wiley & Sons, Ltd.     

Linear and non‐linear stability analysis for finite difference discretizations of high‐order Boussinesq equations

This paper considers a method of lines stability analysis for finite difference discretizations of a recently published Boussinesq method for the study of highly non‐linear and extremely dispersive water waves. The analysis demonstrates the near‐equivalence of classical linear Fourier (von Neumann) techniques with matrix‐based methods for formulations in both one and two horizontal dimensions. The matrix‐based method is also extended to show the local de‐stabilizing effects of the non‐linear terms, as well as the stabilizing effects of numerical dissipation. A comparison of the relative stability of rotational and irrotational formulations in two horizontal dimensions provides evidence that the irrotational formulation has significantly better stability properties when the deep‐water non‐linearity is high, particularly on refined grids. Computation of matrix pseudospectra shows that the system is only moderately non‐normal, suggesting that the eigenvalues are likely suitable for analysis purposes. Numerical experiments demonstrate excellent agreement with the linear analysis, and good qualitative agreement with the local non‐linear analysis. The various methods of analysis combine to provide significant insight into the numerical behaviour of this rather complicated system of non‐linear PDEs. Copyright © 2004 John Wiley & Sons, Ltd.     

Time‐accurate stabilized finite‐element model for weakly nonlinear and weakly dispersive water waves

Introduction of a time‐accurate stabilized finite‐element approximation for the numerical investigation of weakly nonlinear and weakly dispersive water waves is presented in this paper. To make the time approximation match the order of accuracy of the spatial representation of the linear triangular elements by the Galerkin finite‐element method, the fourth‐order time integration of implicit multistage Padé method is used for the development of the numerical scheme. The streamline‐upwind Petrov–Galerkin (SUPG) method with crosswind diffusion is employed to stabilize the scheme and suppress the spurious oscillations, usually common in the numerical computation of convection‐dominated flow problems. The performance of numerical stabilization and accuracy is addressed. Treatments of various boundary conditions, including the open boundary conditions, the perfect reflecting boundary conditions along boundaries with irregular geometry, are also described. Numerical results showing the comparisons with analytical solutions, experimental measurements, and other published numerical results are presented and discussed. Copyright © 2007 John Wiley & Sons, Ltd.