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

Walkley, Mark A.; Berzins, Martin

doi:10.1002/(sici)1097-0363(19990130)29:2<143::aid-fld779>3.0.co;2-5

Cited by 61 publications

(67 citation statements)

References 20 publications

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“…Therefore this experiment has been used as a benchmark test for dispersive wave models. See for example Madsen and Schäffer (1999), Madsen, Bingham and Liu (2002) and Walkley (1999).…”

Section: Harmonic Generation Over a Submerged Barmentioning

confidence: 99%

Nodal DG-FEM solution of high-order Boussinesq-type equations

et al. 2006

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Abstract. We present 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 and two horizontal dimensions. 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 verified 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 a future paper.

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“…Therefore this experiment has been used as a benchmark test for dispersive wave models. See for example Madsen and Schäffer (1999), Madsen, Bingham and Liu (2002) and Walkley (1999).…”

Section: Harmonic Generation Over a Submerged Barmentioning

confidence: 99%

Nodal DG-FEM solution of high-order Boussinesq-type equations

et al. 2006

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“…However, mass lumping decreases the accuracy of the solutions (Walkley and Berzins 1999). As an alternative method to satisfy the needs of both calculation time and accuracy, the Legendre interpolation function proposed by Patera (1984) can be used for the calculation.…”

Section: Formulationmentioning

confidence: 99%

Wave Transformation Modeling with Effective Higher-Order Finite Elements

Jung¹,

Ryu²

2016

JAFM

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This study introduces a finite element method using a higher-order interpolation function for effective simulations of wave transformation. Finite element methods with a higher-order interpolation function usually employ a Lagrangian interpolation function that gives accurate solutions with a lesser number of elements compared to lower order interpolation function. At the same time, it takes a lot of time to get a solution because the size of the local matrix increases resulting in the increase of band width of a global matrix as the order of the interpolation function increases. Mass lumping can reduce computation time by making the local matrix a diagonal form. However, the efficiency is not satisfactory because it requires more elements to get results. In this study, the Legendre cardinal interpolation function, a modified Lagrangian interpolation function, is used for efficient calculation. Diagonal matrix generation by applying direct numerical integration to the Legendre cardinal interpolation function like conducting mass lumping can reduce calculation time with favorable accuracy. Numerical simulations of regular, irregular and solitary waves using the Boussinesq equations through applying the interpolation approaches are carried out to compare the higher-order finite element models on wave transformation and examine the efficiency of calculation.

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“…In general, the presence of third-order spatial derivatives in momentum equation makes the application of a standard linear interpolation FEM very complicated. Walkley and Berzins (1999) offered a new method based on the linear Galerkin FEM to solve the Nwogu Boussinesq equations (NBE). They introduced an auxiliary variable and reformulated the system of equations in its lower-order form.…”

Section: Computational Algorithmmentioning

confidence: 99%

“…Li et al (1999) solved the improved Boussinesq equations by Beji and Nadaoka (1996). The extended Boussinesq equations of Madsen and Sorensen (1992) were solved by Walkley and Berzins (1999), and Walkley (1999). Recently, Sherwin (2002, 2005) solved the same set of extended equations using a Discontinuous Galerkin finite element method along with the classical Boussinesq equations of Peregrine and the extended Boussinesq equations of Madsen and Sorensen.…”

Section: Introductionmentioning

confidence: 99%

Numerical modeling of solitary waves by 1-D Madsen and Sorensen extended Boussinesq equations

Ghadimi

Rahimzadeh

Chekab

et al. 2015

ISH Journal of Hydraulic Engineering

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This paper presents numerical solution of one-dimensional enhanced Boussinesq equations introduced by Madsen and Sorensen for modeling the solitary wave propagation in three different physical situations. Governing equations are spatially discretized using weighted residual Galerkin approach. To implement the linear finite element method, an auxiliary equation is introduced in order to simplify the discretization task of the third-order spatial derivatives present in momentum equation. Exact solitary wave solutions are used to specify the initial data for the incident solitary waves in the numerical model and for verification of the corresponding computed solutions. The proposed model has been used to simulate important wave phenomena including solitary wave propagation, shoaling, absorption and reflection. Validity of the developed code has been assessed by comparing the results against published analytical solutions and experimental measurements. In order to compare the current results against the numerical solution of the extended Boussinesq equation without the auxiliary form, propagation of solitary wave over a shelf is investigated. In all of the considered cases in the current study, the proposed model is shown capable of producing favorable agreements. IntroductionRecent developments in coastal engineering and the presence of barriers such as islands and breakwaters in shallow water depth have heightened the need for examining a realistic modeling of wave transformation in the near shore zone including the non-linear interactions. Such studies can be analyzed experimentally with scale models in wave tanks or numerically by application of a suitable mathematical model and numerical method.Several experiments have been carried out for investigating the evolution of different types of waves propagating in wave tanks in various geometrical settings. For example, Li and Raichlen (2002) studied the run up of a solitary wave in a sloping beach. Two different wave tanks were considered for that test case. In another work, Cooker et al. (1990) investigated the breaking of solitary waves propagating over semicircular bumps in wave tanks. Wu and Hsiao (2013) studied the turbulence induced by a solitary wave propagating over a submerged object by using particle image velocimetry. Particle image velocimetry was also used by Lin et al. (2006) who studied propagation of a solitary wave over a submerged rectangular dike and by Chang et al. (2001Chang et al. ( , 2005 who investigated the vortex generation and propagation of solitary and cnoidal waves over submerged rectangular obstacles. There have been many significant contributions from different experimental works, but experimental studies are relatively expensive and involve further complications. Therefore, in recent years, numerical methods have been widely used for simulating the wave generation.The advantage of numerical models is the ease with which different layouts can be constructed and tested compared to rebuilding of a physical model. However, the assumption...

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

Cited by 61 publications

References 20 publications

Nodal DG-FEM solution of high-order Boussinesq-type equations

Nodal DG-FEM solution of high-order Boussinesq-type equations

Wave Transformation Modeling with Effective Higher-Order Finite Elements

Numerical modeling of solitary waves by 1-D Madsen and Sorensen extended Boussinesq equations

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