A high‐order Petrov–Galerkin finite element method for the classical Boussinesq wave model

A high‐order Petrov–Galerkin finite element scheme is presented to solve the one‐dimensional depth‐integrated classical Boussinesq equations for weakly non‐linear and weakly dispersive waves. Finite elements are used both in the space and the time domains. The shape functions are bilinear in space–time, whereas the weighting functions are linear in space and quadratic in time, with C⁰‐continuity. Dispersion correction and a highly selective dissipation mechanism are introduced through additional streamline upwind terms in the weighting functions. An implicit, conditionally stable, one‐step predictor–corrector time integration scheme results. The accuracy and stability of the non‐linear discrete equations are investigated by means of a local Taylor series expansion. A linear spectral analysis is used for the full characterization of the predictor–corrector inner iterations. Based on the order of the analytical terms of the Boussinesq model and on the order of the numerical discretization, it is concluded that the scheme is fourth‐order accurate in terms of phase velocity. The dissipation term is third order only affecting the shortest wavelengths. A numerical convergence analysis showed a second‐order convergence rate in terms of both element size and time step. Four numerical experiments are addressed and their results are compared with analytical solutions or experimental data available in the literature: the propagation of a solitary wave, the oscillation of a flat bottom closed basin, the oscillation of a non‐flat bottom closed basin, and the propagation of a periodic wave over a submerged bar. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Numerical solutions of fully non‐linear and highly dispersive Boussinesq equations in two horizontal dimensions

This paper investigates preconditioned iterative techniques for finite difference solutions of a high‐order Boussinesq method for modelling water waves in two horizontal dimensions. The Boussinesq method solves simultaneously for all three components of velocity at an arbitrary z‐level, removing any practical limitations based on the relative water depth. High‐order finite difference approximations are shown to be more efficient than low‐order approximations (for a given accuracy), despite the additional overhead. The resultant system of equations requires that a sparse, unsymmetric, and often ill‐conditioned matrix be solved at each stage evaluation within a simulation. Various preconditioning strategies are investigated, including full factorizations of the linearized matrix, ILU factorizations, a matrix‐free (Fourier space) method, and an approximate Schur complement approach. A detailed comparison of the methods is given for both rotational and irrotational formulations, and the strengths and limitations of each are discussed. Mesh‐independent convergence is demonstrated with many of the preconditioners for solutions of the irrotational formulation, and solutions using the Fourier space and approximate Schur complement preconditioners are shown to require an overall computational effort that scales linearly with problem size (for large problems). Calculations on a variable depth problem are also compared to experimental data, highlighting the accuracy of the model. Through combined physical and mathematical insight effective preconditioned iterative solutions are achieved for the full physical application range of the model. Copyright © 2004 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.     

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.     

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.     

Oscillations in high‐order finite difference solutions of stiff problems on non‐uniform grids

This work investigates the mitigation and elimination of scheme‐related oscillations generated in compact and classical fourth‐order finite difference solutions of stiff problems, represented here by the Burgers and Reynolds equations. The regions where severe gradients are anticipated are refined by the use of subdomains where the grid is distributed according to a geometric progression. It is observed that, for multi‐domain solutions, both the classical and compact fourth‐order finite difference schemes can exhibit spurious oscillations. When present, the oscillations are initially generated around the interface between the uniform and non‐uniform grid subdomains. Based on a thorough study of the grid distribution effects, it is shown that the numerical oscillations are caused by inadequate geometric progression ratios within the non‐uniformly discretized subdomains. Indeed, accurate solutions are obtainable if and only if the grid ratios in the non‐uniform subdomains are greater than a critical threshold ratio. It is concluded that high‐order classical and compact schemes can be used with confidence to efficiently solve one‐ or two‐dimensional problems whose solutions exhibit sharp gradients in very thin regions, provided that the numerically generated oscillations are eliminated by an appropriate choice of grid distribution within the non‐uniformly discretized subdomains. Copyright © 1999 John Wiley & Sons, Ltd.     

A Petrov–Galerkin finite element model for one‐dimensional fully non‐linear and weakly dispersive wave propagation

A new finite element method is presented to solve one‐dimensional depth‐integrated equations for fully non‐linear and weakly dispersive waves. For spatial integration, the Petrov–Galerkin weighted residual method is used. The weak forms of the governing equations are arranged in such a way that the shape functions can be piecewise linear, while the weighting functions are piecewise cubic with C²‐continuity. For the time integration an implicit predictor–corrector iterative scheme is employed. Within the framework of linear theory, the accuracy of the scheme is discussed by considering the truncation error at a node. The leading truncation error is fourth‐order in terms of element size. Numerical stability of the scheme is also investigated. If the Courant number is less than 0.5, the scheme is unconditionally stable. By increasing the number of iterations and/or decreasing the element size, the stability characteristics are improved significantly. Both Dirichlet boundary condition (for incident waves) and Neumann boundary condition (for a reflecting wall) are implemented. Several examples are presented to demonstrate the range of applicabilities and the accuracy of the model. Copyright © 2001 John Wiley & Sons, Ltd.     

A non‐linear coupled fluid–structure aeroelastic analysis of a slender bridge deck

This paper presents a new numerical methodology for the integral aeroelastic instability analysis of slender structures, based on the appropriate conjugation of an algorithm for dynamic and geometrically non‐linear analysis of structures based on the finite element method with another algorithm of computational fluid dynamics (Finite volume method). It is considered a viscous incompressible unsteady turbulent bidimensional air flow solved on a structured control volume mesh. The computer code developed on the basis of this methodology is applied to the aeroelastic study of a simply supported slender bridge deck in order to find out the critical wind velocity leading to instability. Some of the most significant results associated with the analysis of the corresponding aeroelastic behaviour are presented. Copyright 2004 John Wiley & Sons, Ltd.     

Solving a fully nonlinear highly dispersive Boussinesq model with mesh‐less least square‐based finite difference method

Combining mesh‐less finite difference method and least square approximation, a new numerical model is developed for water wave propagation model in two horizontal dimensions. In the numerical formulation of the method, the approximation of the unknown functions and their derivatives are constructed on a set of nodes in a local circular‐shaped region. The Boussinesq equations studied in this paper is a fully nonlinear and highly dispersive model, which is composed of the exact boundary conditions and the truncated series expansion solution of the Laplace equation. The resultant system involves a sparse, unsymmetrical matrix to be solved at each time step of the simulation. Matrix solutions are studied to reduce the computing resource requirements and improve the efficiency and accuracy. The convergence properties of the present numerical method are investigated. Preliminary verifications are given for nonlinear wave shoaling problems; the numerical results agree well with experimental data available in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

Global linear stability analysis of time‐averaged flows

The global linear stability analysis (LSA) of stationary/steady flows has been applied to various flows in the past and is fairly well understood. The LSA of time‐averaged flows is explored in this paper. It is shown that the LSA of time‐averaged flows can result in useful information regarding its stability. The method is applied to study flow past a cylinder at Reynolds number (Re) beyond the onset of vortex shedding. Compared with the direct numerical simulation, LSA of the Re=100 steady flow severely underpredicts the vortex shedding frequency. However, the LSA of the time‐averaged flow results in the correct value of the non‐dimensional frequency, St, of the associated instability. Copyright © 2007 John Wiley & Sons, Ltd.     

Local maximum principle satisfying high‐order non‐oscillatory schemes

The main contribution of this work is to classify the solution region including data extrema for which high‐order non‐oscillatory approximation can be achieved. It is performed in the framework of local maximum principle (LMP) and non‐conservative formulation. The representative uniformly second‐order accurate schemes are converted in to their non‐conservative form using the ratio of consecutive gradients. Using the local maximum principle, these non‐conservative schemes are analyzed for their non‐linear LMP/total variation diminishing stability bounds which classify the solution region where high‐order accuracy can be achieved. Based on the bounds, second‐order accurate hybrid numerical schemes are constructed using a shock detector. The presented numerical results show that these hybrid schemes preserve high accuracy at non‐sonic extrema without exhibiting any induced local oscillations or clipping error. Copyright © 2015 John Wiley & Sons, Ltd.     

A fourth‐order compact finite difference scheme for the steady stream function–vorticity formulation of the Navier–Stokes/Boussinesq equations

A fourth‐order compact finite difference scheme on the nine‐point 2D stencil is formulated for solving the steady‐state Navier–Stokes/Boussinesq equations for two‐dimensional, incompressible fluid flow and heat transfer using the stream function–vorticity formulation. The main feature of the new fourth‐order compact scheme is that it allows point‐successive overrelaxation (SOR) or point‐successive underrelaxation iteration for all Rayleigh numbers Ra of physical interest and all Prandtl numbers Pr attempted. Numerical solutions are obtained for the model problem of natural convection in a square cavity with benchmark solutions and compared with some of the accurate results available in the literature. Copyright © 2003 John Wiley & Sons, Ltd.     

A finite element formulation for global linear stability analysis of a nominally two‐dimensional base flow

A stabilized finite element method, to carry out the linear stability analysis of a two‐dimensional base flow to three‐dimensional perturbations that are periodic along span, is presented. The resulting equations for the time evolution of the disturbance requires a solution to the generalized eigenvalue problem. The analysis is global in nature and is also applicable to non‐parallel flows. Equal‐order‐interpolation functions for velocity and pressure are utilized. Stabilization terms are added to the Galerkin formulation to admit the use of equal‐order‐interpolation functions and to eliminate node‐to‐node oscillations that might arise in advection‐dominated flows. The proposed formulation is tested on two flow problems. First, the mode transitions in the circular Couette flow are investigated. Two scenarios are considered. In the first one, the outer cylinder is at rest, while the inner one spins. Two linearly unstable modes are identified. The primary mode is real and represents the axisymmetric Taylor vortices. The second mode is complex and consists of spiral vortices. For the counter‐rotating cylinders, the primary transition is via the appearance of spiral vortices. Excellent agreement with results from earlier studies is observed. The formulation is also utilized to investigate the parallel and oblique modes of vortex shedding past a cylinder for the Re = 100 flow. It is found that the flow is associated with a large number of unstable oblique shedding modes. The parallel mode of vortex shedding is a special case of this family of modes and is associated with the largest growth rate. Copyright © 2014 John Wiley & Sons, Ltd.     

Hybrid finite‐volume finite‐difference scheme for the solution of Boussinesq equations

A hybrid scheme composed of finite‐volume and finite‐difference methods is introduced for the solution of the Boussinesq equations. While the finite‐volume method with a Riemann solver is applied to the conservative part of the equations, the higher‐order Boussinesq terms are discretized using the finite‐difference scheme. Fourth‐order accuracy in space for the finite‐volume solution is achieved using the MUSCL‐TVD scheme. Within this, four limiters have been tested, of which van‐Leer limiter is found to be the most suitable. The Adams–Basforth third‐order predictor and Adams–Moulton fourth‐order corrector methods are used to obtain fourth‐order accuracy in time. A recently introduced surface gradient technique is employed for the treatment of the bottom slope. A new model ‘HYWAVE’, based on this hybrid solution, has been applied to a number of wave propagation examples, most of which are taken from previous studies. Examples include sinusoidal waves and bi‐chromatic wave propagation in deep water, sinusoidal wave propagation in shallow water and sinusoidal wave propagation from deep to shallow water demonstrating the linear shoaling properties of the model. Finally, sinusoidal wave propagation over a bar is simulated. The results are in good agreement with the theoretical expectations and published experimental results. Copyright © 2005 John Wiley & Sons, Ltd.