Energy-preserving finite volume element method for the improved Boussinesq equation

Wang, Quanxiang; Zhang, Zhiyue; Zhang, Xinhua; Zhu, Quanyong

doi:10.1016/j.jcp.2014.03.053

Cited by 52 publications

(46 citation statements)

References 22 publications

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“…For these issues, we refer to [47,2,57] and references therein, concerning the PDE continuous modelling, and to [76,26,91] and references therein, for aspects related to the energy stable approximation of the shallow water equations. The construction of exactly energy preserving schemes for dispersive equations is still a subject of research, the interested reader may refer to [84,35,93,92] for some recent results concerning dispersive equations.…”

Section: Breaking Bore Propagation and Energy Dissipationmentioning

confidence: 99%

On wave breaking for Boussinesq-type models

Καζολέα¹,

Ricchiuto²

2018

Ocean Modelling

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Abstract:We considers the issue of wave breaking closure for Boussinesq type models, and attempt at providing some more understanding of the sensitivity of some closure approaches to the numerical setup. In particular, we are interested in the potential for mesh refinement, and in quantifying the dissipation mechanism active. Two closure strategies are considered. The first is an eddy viscosity approach following some early work by O. Nwogu in the 90's, and some more recent developments by Zhang and co-workers. In this model, a breaking viscosity is computed starting form a turbulent kinetic energy. The latter is obtained from an ad-hoc partial differential equation, which is solved in parallel with the propagation model. The second approach considered consists in suppressing the dispersive terms in breaking regions. The dissipation of total energy obtained in a shallow water shock is used to model the energy dissipation due to breaking. Due to its simplicity and effectiveness, the second approach has gained substantial attention in the coastal engineering community. Here we propose a systematic comparison of the two approaches, to understand more of their sensitivity w.r.t. the type of propagation model used (weakly or fully nonlinear), to the type of breaking wave being simulated, as well as to understand the mesh dependence of the pointwise results obtained, and in particular the potential for achieving mesh converged simulations. Finally, we provide a quantitative analysis of the dissipation introduced by the two closures for a moving breaking bore.

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Section: Breaking Bore Propagation and Energy Dissipationmentioning

confidence: 99%

On wave breaking for Boussinesq-type models

Καζολέα¹,

Ricchiuto²

2018

Ocean Modelling

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“…In this subsection, we consider the improved Boussinesq Equation 1.5 on the domain

[- 80, 80]

with the following exact solution :

u (x, t) = A sec h^{2} (\sqrt{\frac{A}{6}} \frac{x - x_{0} - β t}{normalβ}), β = \sqrt{1 + \frac{2 A}{3}},

where

A

x_{0}

and

normalβ

are given constants. The exact solution represents a solitary wave with amplitude

A

, and initially located at

x = x_{0}

and moved to the right or left (depending on the sign of

normalβ

) with velocity

normalβ

.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…The exact solution represents a solitary wave with amplitude

A

, and initially located at

x = x_{0}

and moved to the right or left (depending on the sign of

normalβ

) with velocity

normalβ

. This example is given in . The initial conditions can be obtained from the exact solution:

\begin{array}{l} left & u (x, 0) = A sec h^{2} (\sqrt{\frac{A}{6}} (x - x_{0})), \\ left & u_{t} (x, 0) = 2 A \sqrt{\frac{A}{6}} sec h^{2} (\sqrt{\frac{A}{6}} \frac{x - x_{0}}{normalβ}) \tanh (\sqrt{\frac{A}{6}} \frac{x - x_{0}}{normalβ}) . \end{array}

To estimate the accuracy and the conservation properties, we fix

A = .5

x_{0} = 0

β = \sqrt{1 + 2 A / 3}

and integrate up to time

T = 1

.…”

Section: Numerical Experimentsmentioning

confidence: 99%

High‐order energy‐preserving schemes for the improved Boussinesq equation

Yan

Zhang

Zhao

et al. 2018

Numerical Methods Partial

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This article proposes a class of high‐order energy‐preserving schemes for the improved Boussinesq equation. To derive the energy‐preserving schemes, we first discretize the improved Boussinesq equation by Fourier pseudospectral method, which leads to a finite‐dimensional Hamiltonian system. Then, the obtained semidiscrete system is solved by Hamiltonian boundary value methods, which is a newly developed class of energy‐preserving methods. The proposed schemes can reach spectral precision in space, and in time can reach second‐order, fourth‐order, and sixth‐order accuracy, respectively. Moreover, the proposed schemes can conserve the discrete mass and energy to within machine precision. Furthermore, to show the efficiency and accuracy of the proposed methods, the proposed methods are compared with the finite difference methods and the finite volume element method. The results of several numerical experiments are given for the propagation of the single solitary wave, the interaction of two solitary waves and the wave break‐up.

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“…Yan et al (2016) reported a two-grid finite volume element scheme for nonlinear Sobolev equations. Wang et al (2014) proposed an energy-preserving finite volume element scheme for the improved Boussinesq equation. In this paper, based on the DVDM and the FVEM, we develop an energy-preserving scheme which is accurate, unconditionally stable (with a long time computation ability) to solve the CMKDV numerically.…”

Section: Introductionmentioning

confidence: 99%

Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

Yan

Zheng

2017

International Journal of Applied Mathematics and Computer Science

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The aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg-de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg-de Vries equation.

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Energy-preserving finite volume element method for the improved Boussinesq equation

Cited by 52 publications

References 22 publications

On wave breaking for Boussinesq-type models

On wave breaking for Boussinesq-type models

High‐order energy‐preserving schemes for the improved Boussinesq equation

Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

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