Boussinesq-Peregrine water wave models and their numerical approximation (original) (raw)
Related papers
Dynamics of modified improved Boussinesq equation via Galerkin Finite Element Method
Mathematical Methods in the Applied Sciences, 2020
The aim of this paper is to investigate numerical solutions of modified improved Boussinesq (MIBq) equation u tt = u xx + α u 3 ð Þ xx + u xxtt , which is a modified type of Boussinesq equations born as an art of modelling water-wave problems in weakly dispersive medium such as surface waves in shallow waters or ion acoustic waves. For this purpose, Lumped Galerkin finite element (LGFE) method, an effective, accurate, and cost-effective method, is applied to model equation by the aid of quadratic B-spline basis. The efficiency and accuracy of the method are tested with two problems, namely, propagation solitary wave and interaction of two solitary waves. The error norms L 2 and L ∞ have been computed in order to measure how "accurate" the numerical solutions. Also, the stability analysis has been investigated.
A high-order Petrov-Galerkin finite element method for the classical Boussinesq wave model
International Journal for Numerical Methods in Fluids, 2009
A high-order Petrov-Galerkin finite element scheme is presented to solve the one-dimensional depthintegrated 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 spacetime, whereas the weighting functions are linear in space and quadratic in time, with C 0-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 predictorcorrector 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.
Numerical modeling of solitary waves by 1-D Madsen and Sorensen extended Boussinesq equations
ISH Journal of Hydraulic Engineering, 2015
In this paper, finite element modeling of one-dimensional extended Boussinesq equations derived by Madsen and Sorensen is presented for simulation of propagating regular waves. In order to spatially discretize the finite element equations, method of weighted residual Galerkin approach is used. Discretization of third-order derivative in momentum equation is performed by introducing of an auxiliary equation, which makes it possible to use linear finite element method. Adams-Bashforth-Moulton predictor-corrector method is used for time integration. Regular wave trains are simulated using the proposed numerical scheme. For validation of the developed code, the model is applied to several examples of wave propagation over the computational domain and the obtained results of the current computations are compared against the experimental measurements. In all cases, the proposed model has proved very suitable for simulating the propagation of wave indicating favorable agreements with experimental data.
Finite Volume Solution of Boussinesq-Type Equations on an Unstructured Grid
2007
A new numerical method is developed to solve a set of two-dimensional Boussinesq water wave evolution equations over an unstructured grid. The governing mass and momentum conservation equations are discretized over an irregular triangular grid, with a staggered placement of the variables. The free surface elevation is defined at the centroid of the triangles, while the normal component of the velocity is defined at the mid-point of the triangle edges. The mass conservation equation is then integrated over a control volume defined over each triangle while the momentum equations are integrated over a control volume formed from two adjacent triangles. A modified Crank-Nicolson scheme is used to integrate the equations in time. Two numerical experiments are used to evaluate the conservation properties and accuracy of the numerical method: solitary wave propagation in a curved channel, and interaction of solitary waves with a vertical circular cylinder.
On the Galerkin/Finite-Element Method for the Serre Equations
Journal of Scientific Computing, 2014
A highly accurate numerical scheme is presented for the Serre system of partial differential equations, which models the propagation of dispersive shallow water waves in the fully-nonlinear regime. The fully-discrete scheme utilizes the Galerkin / finite-element method based on smooth periodic splines in space, and an explicit fourth-order Runge-Kutta method in time. Computations compared with exact solitary and cnoidal wave solutions show that the scheme achieves the optimal orders of accuracy in space and time. These computations also show that the stability of this scheme does not impose very restrictive conditions on the temporal stepsize. In addition, solitary, cnoidal, and dispersive shock waves are studied in detail using this numerical scheme for the Serre system and compared with the 'classical' Boussinesq system for small-amplitude shallow water waves. The results show that the interaction of solitary waves in the Serre system is more inelastic. The efficacy of the numerical scheme for modeling dispersive shocks is shown by comparison with asymptotic results. These results have application to the modeling of shallow water waves of intermediate or large amplitude.
Nodal DG-FEM solution of high-order Boussinesq-type equations
Journal of Engineering Mathematics, 2007
A discontinuous Galerkin finite-element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one horizontal dimension is presented. The continuous equations are discretized using nodal polynomial basis functions of arbitrary order in space on each element of an unstructured computational domain. A fourth-order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy and convergence of the model with both h (grid size) and p (order) refinement are confirmed for the linearized equations, and calculations are provided for two nonlinear test cases in one horizontal dimension: harmonic generation over a submerged bar, and reflection of a steep solitary wave from a vertical wall. Test cases for two horizontal dimensions will be considered in future work.
Journal of Computational Physics, 2014
A local discontinuous Galerkin method for Boussinesq-Green Naghdi Equations is presented and validated against experimental results for wave transformation over a submerged shoal. Currently Green-Naghdi equations have many variants. In this paper a numerical method in one dimension is presented for the Green -Naghdi equations based on rotational characteristics in the velocity field. Stability criterion is also established for the linearized Green-Naghdi equations for both the analytical problem and the numerical method. Verification is done against a linearized standing wave problem in flat bathymetry and h,p (denoted by K in this paper) error rates are plotted. Validation plots show good agreement of the numerical results with the experimental ones.
Journal of Fluid Mechanics, 2003
The objective of this paper is to discuss and analyse the accuracy of various velocity formulations for water waves in the framework of Boussinesq theory. To simplify the discussion, we consider the linearized wave problem confined between the stillwater datum and a horizontal sea bottom. First, the problem is further simplified by ignoring boundary conditions at the surface. This reduces the problem to finding truncated series solutions to the Laplace equation with a kinematic condition at the sea bed. The convergence and accuracy of the resulting expressions is analysed in comparison with the target cosh-and sinh-functions from linear wave theory. First, we consider series expansions in terms of the horizontal velocity variable at an arbitrary z-level, which can be varied from the sea bottom to the still-water datum. Second, we consider the classical possibility of expanding in terms of the depth-averaged velocity. Third, we analyse the use of a horizontal pseudo-velocity determined by interpolation between velocities at two arbitrary z-levels. Fourth, we investigate three different formulations based on two expansion variables, being the horizontal and vertical velocity variables at an arbitrary z-level. This is shown to have a remarkable influence on the convergence and to improve accuracy considerably. Fifth, we derive and analyse a new formulation which doubles the power of the vertical coordinate without increasing the order of the horizontal derivatives. Finally, we involve the kinematic and dynamic boundary conditions at the free surface and discuss the linear dispersion relation and a spectral solution for steady nonlinear waves.
Journal of Computational Physics, 2014
A new methodology is presented to handle wave breaking over complex bathymetries in extended two-dimensional Boussinesq-type (BT) models which are solved by an unstructured well-balanced finite volume (FV) scheme. The numerical model solves the 2D extended BT equations proposed by Nwogu (1993), recast in conservation law form with a hyperbolic flux identical to that of the Non-linear Shallow Water (NSW) equations. Certain criteria, along with their proper implementation, are established to characterize breaking waves. Once breaking waves are recognized, we switch locally in the computational domain from the BT to NSW equations by suppressing the dispersive terms in the vicinity of the wave fronts. Thus, the shock-capturing features of the FV scheme enable an intrinsic representation of the breaking waves, which are handled as shocks by the NSW equations. An additional methodology is presented on how to perform a stable switching between the BT and NSW equations within the unstructured FV framework. Extensive validations are presented, demonstrating the performance of the proposed wave breaking treatment, along with some comparisons with other well-established wave breaking mechanisms that have been proposed for BT models.
DG-FEM solution for nonlinear wave-structure interaction using Boussinesq-type equations
Coastal Engineering, 2008
We present a high-order nodal Discontinuous Galerkin Finite Element Method (DG-FEM) solution based on a set of highly accurate Boussinesq-type equations for solving general water-wave problems in complex geometries. A nodal DG-FEM is used for the spatial discretization to solve the Boussinesq equations in complex and curvilinear geometries which amends the application range of previous numerical models that have been based on structured Cartesian grids. The Boussinesq method provides the basis for the accurate description of fully nonlinear and dispersive water waves in both shallow and deep waters within the breaking limit. To demonstrate the current applicability of the model both linear and nonlinear test cases are considered where the water waves interact with bottom-mounted fully reflecting structures. It is established that, by simple symmetry considerations combined with a mirror principle, it is possible to impose weak slip boundary conditions for both structured and general curvilinear wall boundaries while maintaining the accuracy of the scheme. As is standard for current high-order Boussinesq-type models, arbitrary waves can be generated and absorbed in the interior of the computational domain using a flexible relaxation technique applied on the free surface variables.