Numerical solution of Boussinesq systems of the Bona–Smith family

Antonopoulos, D. C.; Dougalis, Vassilios A.; Mitsotakis, Dimitrios

doi:10.1016/j.apnum.2009.03.002

Cited by 48 publications

(88 citation statements)

References 58 publications

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“…Overtaking collision has been studied recently in the case of bidirectional models in [4]. The interaction is similar to that of the unidirectional models but it was found that a new N-shape wavelet is generated during the interaction.…”

Section: Overtaking Collisionsmentioning

confidence: 93%

“…These effects have been studied extensively before by numerical means using high order numerical methods such as finite differences, [11], spectral and finite element methods [4,27,62] and experimentally in [22]. In Fig.…”

Section: Head-on Collisionsmentioning

confidence: 99%

“…After the collision we observe that the phase shift of the solitary waves is the same in both numerical and experimental data, while the shape of the experimental solitary waves were not stabilized due to interactions with other small amplitude dispersive waves. We note that after the head-on collision of the waves small amplitude dispersive tails were developed, [4,11,27].…”

Section: Head-on Collisionsmentioning

confidence: 99%

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Finite volume schemes for dispersive wave propagation and runup

Dutykh

Katsaounis

Mitsotakis

2011

Journal of Computational Physics

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Finite volume schemes are commonly used to construct approximate solutions to conservation laws. In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems. We focus mainly on the application of the method to bidirectional nonlinear, dispersive wave propagation in one space dimension. Special emphasis is given to important nonlinear phenomena such as solitary waves interactions, dispersive shock wave formation and the runup of breaking and non-breaking long waves.Comment: 41 pafes, 20 figures. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

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Section: Overtaking Collisionsmentioning

confidence: 93%

Section: Head-on Collisionsmentioning

confidence: 99%

Section: Head-on Collisionsmentioning

confidence: 99%

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Finite volume schemes for dispersive wave propagation and runup

Dutykh

Katsaounis

Mitsotakis

2011

Journal of Computational Physics

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“…The travelling waves for the KdV-KdV systems were also studied numerically by Bona et al using unconditionally stable periodic splines Galerkin method with Gauss-Legendre implicit Runge-Kutta method of stage two [5]. Antonopoulos et al simulated the numerical solutions of three initial boundary value problems one parameter Bona-Smith systems [6]. The problems with homogenous Dirichlet, reflection, and periodic boundary conditions were integrated by Galerkin-finite element combined with fourth order explicit Runge-Kutta method.…”

Section: Boussinesq Systemsmentioning

confidence: 99%

Exponential B-Splines for Numerical Solutions to Some Boussinesq Systems for Water Waves

2016

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In the study, the collocation method based on exponential cubic B-spline functions is proposed to solve one dimensional Boussinesq systems numerically. Two initial boundary value problems for Regularized and Classical Boussinesq systems modeling motion of traveling waves are considered. The accuracy of the method is validated by measuring the error between the numerical and analytical solutions. The numerical solutions obtained by various values of free parameter p are compared with some solutions in literature.Keywords: Boussinesq sytems; solitary waves; exponential cubic B-spline; collocation. MSC2010: 35Q99;35C07;76B25;65D07;65M70. Boussinesq SystemsIn the light of the d' Alembert solution, describing two distinct waves moving in the opposite directions for the Cauchy problem for one dimensional wave equation, many physical problems modeled by linear and nonlinear partial differential equations have been solved in wave forms covering traveling waves, solitary waves, harmonic waves, etc. Waves are an important solution class for the model problems in physics, chemistry, and many fields of engineering. This solution type is widely constructed for different models in various media covering solids and fluids. Class of surface bell-shaped solitary waves, having long amplitude when compared with its width, is one of the most prominent classes of waves. In addition to linear equations, solitary wave solutions have also been studied for well known nonlinear equations and systems such as Korteweg-deVries(KdV), Schrödinger, Boussinesq, etc. Having analytical solutions only for some particular cases including additional restrictions and conditions, many numerical methods have been developed to obtain solutions for nonlinear problems. Dougalis et al.[1] studied the numerical behavior of solitary waves of a member of Boussinesq systems family (Bona-Smith system). In that study, long time stability and possible blow-up solutions were examined under small and large perturbations covering perturbation of amplitudes, system coefficients, etc. Numerical models were constructed with the Galerkin-finite element method on the space S h of smooth, periodic, cubic splines. Quintic B-spline collocation tecnique based on Crank-Nicolson formulation was set up for numerical solutions of Boussines type coupled-BBM system [2]. Two initial boundary value problems describing motion of single solitary wave and interaction of solitary waves were simulated by the proposed method. The numerical results were compared with the analytical ones. * Ozlem Ersoy,

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“…A pseudo-spectral method was applied in [25], an implicit finite difference scheme in [53,7] and a compact higher-order scheme in [16,17]. Some Galerkin and Finite Element type methods have been successfully applied to Boussinesq-type equations [27,54,4,3]. A finite difference discretization based on an integral formulation was proposed by Bona & Chen [10].…”

Section: Introductionmentioning

confidence: 99%

Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations

et al. 2013

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Abstract. After we derive the Serre system of equations of water wave theory from a generalized variational principle, we present some of its structural properties. We also propose a robust and accurate finite volume scheme to solve these equations in one horizontal dimension. The numerical discretization is validated by comparisons with analytical, experimental data or other numerical solutions obtained by a highly accurate pseudo-spectral method.

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Numerical solution of Boussinesq systems of the Bona–Smith family

Cited by 48 publications

References 58 publications

Finite volume schemes for dispersive wave propagation and runup

Finite volume schemes for dispersive wave propagation and runup

Exponential B-Splines for Numerical Solutions to Some Boussinesq Systems for Water Waves

Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations

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