scholarly journals NUMERICAL STUDY OF ROSENAU-KDV EQUATION USING FINITE ELEMENT METHOD BASED ON COLLOCATION APPROACH

2017 ◽  
Vol 22 (3) ◽  
pp. 373-388 ◽  
Author(s):  
Turgut Ak ◽  
Sharanjeet Dhawan ◽  
S. Battal Gazi Karakoc ◽  
Samir K. Bhowmik ◽  
Kamal R. Raslan
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Numerical Study ◽  
Kdv Equation ◽  
Uniform Mesh ◽  
Shallow Water Waves ◽  
Undular Bore ◽  
Von Neumann ◽  
B Spline ◽  
Collocation Approach

In the present paper, a numerical method is proposed for the numerical solution of Rosenau-KdV equation with appropriate initial and boundary conditions by using collocation method with septic B-spline functions on the uniform mesh points. The method is shown to be unconditionally stable using von-Neumann technique. To check accuracy of the error norms L2 and L∞ are computed. Interaction of two and three solitary waves are used to discuss the effect of the behavior of the solitary waves during the interaction. Furthermore, evolution of solitons is illustrated by undular bore initial condition. These results show that the technique introduced here is suitable to investigate behaviors of shallow water waves.

Numerical Treatment for the Coupled-BBM System

2016 ◽  
Vol 7 (2) ◽  
pp. 67 ◽  
Author(s):  
Khalid K. Ali ◽  
K. R. Raslan ◽  
Talaat S. EL-Danaf
Keyword(s):  
Boundary Conditions ◽  
Numerical Solution ◽  
Solitary Waves ◽  
Nonlinear Term ◽  
Uniform Mesh ◽  
Test Accuracy ◽  
Numerical Treatment ◽  
Unconditionally Stable ◽  
Von Neumann ◽  
B Spline

// In the present paper, a numerical method is proposed for the numerical solution of a coupled-BBM system with appropriate initial and boundary conditions by using collocation method with quintic B-spline on the uniform mesh points. The method is shown to be unconditionally stable using von-Neumann technique. To test accuracy the error norms \(L_2\), \(L_\infity\) are computed. Furthermore, interaction of two and three solitary waves are used to discuss the effect of the behavior of the solitary waves after the interaction. These results show that the technique introduced here is easy to apply. We make linearization for the nonlinear term.

scholarly journals Collocation method with quintic b-spline method for solving hirota-satsuma coupled KDV equation

2016 ◽  
Vol 5 (2) ◽  
pp. 123 ◽  
Author(s):  
K. R. Raslan ◽  
Talaat S. El-Danaf ◽  
Khalid K. Ali
Keyword(s):  
Collocation Method ◽  
Nonlinear Term ◽  
Kdv Equation ◽  
Coupled System ◽  
Uniform Mesh ◽  
Test Accuracy ◽  
Spline Method ◽  
Von Neumann ◽  
B Spline ◽  
Invariants Of Motion

In the present paper, a numerical method is proposed for the numerical solution of a coupled system of KdV (CKdV) equation with appropriate initial and boundary conditions by using collocation method with quintic B-spline on the uniform mesh points. The method is shown to be unconditionally stable using von-Neumann technique. To test accuracy the error norms, are computed. Three invariants of motion are predestined to determine the preservation properties of the problem, and the numerical scheme leads to careful and active results. Furthermore, interaction of two and three solitary waves is shown. These results show that the technique introduced here is easy to apply. We make linearization for the nonlinear term.

Integrable Models

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Fluid Mechanics ◽  
Soliton Solution ◽  
Water Waves ◽  
Heisenberg Model ◽  
Kdv Equation ◽  
Short Review ◽  
Lax Pair ◽  
Shallow Water Waves ◽  
Initial Value ◽  
The Kdv Equation

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

Asynchronous whirl in a rotating cylinder partially filled with liquid

1985 ◽  
Vol 150 ◽  
pp. 311-327 ◽  
Author(s):  
A. S. Berman ◽  
T. S. Lundgren ◽  
A. Cheng
Keyword(s):  
Surface Waves ◽  
Shallow Water ◽  
Solitary Waves ◽  
Water Waves ◽  
Rotating Cylinder ◽  
The Self ◽  
Hydraulic Jumps ◽  
Centrifugal Forces ◽  
Shallow Water Waves ◽  
Analytical Results

Experimental and analytical results are presented for the self-excited oscillations that occur in a partially filled centrifuge when centrifugal forces interact with shallow-water waves. Periodic and aperiodic modulations of the basic whirl phenomena are both observed and calculated. The surface waves are found to be hydraulic jumps, undular bores or solitary waves.

scholarly journals Interaction of linear modulated waves and unsteady dispersive hydrodynamic states with application to shallow water waves

2019 ◽  
Vol 875 ◽  
pp. 1145-1174 ◽  
Author(s):  
T. Congy ◽  
G. A. El ◽  
M. A. Hoefer
Keyword(s):  
Water Waves ◽  
Large Scale ◽  
Kdv Equation ◽  
Linear Wave ◽  
Mean Flow ◽  
Undular Bore ◽  
Expansion Waves ◽  
Flow Interaction ◽  
The Kdv Equation ◽  
New Type

A new type of wave–mean flow interaction is identified and studied in which a small-amplitude, linear, dispersive modulated wave propagates through an evolving, nonlinear, large-scale fluid state such as an expansion (rarefaction) wave or a dispersive shock wave (undular bore). The Korteweg–de Vries (KdV) equation is considered as a prototypical example of dynamic wavepacket–mean flow interaction. Modulation equations are derived for the coupling between linear wave modulations and a nonlinear mean flow. These equations admit a particular class of solutions that describe the transmission or trapping of a linear wavepacket by an unsteady hydrodynamic state. Two adiabatic invariants of motion are identified that determine the transmission, trapping conditions and show that wavepackets incident upon smooth expansion waves or compressive, rapidly oscillating dispersive shock waves exhibit so-called hydrodynamic reciprocity recently described in Maiden et al. (Phys. Rev. Lett., vol. 120, 2018, 144101) in the context of hydrodynamic soliton tunnelling. The modulation theory results are in excellent agreement with direct numerical simulations of full KdV dynamics. The integrability of the KdV equation is not invoked so these results can be extended to other nonlinear dispersive fluid mechanic models.

Experiments on strong interactions between solitary waves

1978 ◽  
Vol 85 (3) ◽  
pp. 417-431 ◽  
Author(s):  
P. D. Weidman ◽  
T. Maxworthy
Keyword(s):  
Shallow Water ◽  
Solitary Waves ◽  
Water Waves ◽  
Vries Equation ◽  
Rectangular Channel ◽  
Quantitative Agreement ◽  
Strong Interactions ◽  
Shallow Water Waves ◽  
Qualitative And Quantitative ◽  
The Waves

Experiments on the interaction between solitary shallow-water waves propagating in the same direction have been performed in a rectangular channel. Two methods were devised to compensate for the dissipation of the waves in order to compare results with Hirota's (1971) solution for the collision of solitons described by the Kortewegde Vries equation. Both qualitative and quantitative agreement with theory is obtained using the proposed corrections for wave damping.

Manoeuvring Study of a Container Ship in Shallow Water Waves

2018 ◽  
Author(s):  
Manases Tello Ruiz ◽  
Marc Mansuy ◽  
Guillaume Delefortrie ◽  
Marc Vantorre
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Numerical Study ◽  
Scale Model ◽  
Wave Forces ◽  
Shallow Water Waves ◽  
The North ◽  
Captive Model Tests ◽  
Calm Water ◽  
Large Container

When approaching or leaving a port a ship often needs to perform manoeuvres in the presence of waves. At the same time the water depth is still limited for deep drafted vessels. For manoeuvring simulation purposes this requires a manoeuvring model which includes phenomena such as short crested waves and squat effects. The present paper addresses the manoeuvring problem in shallow water waves numerically and experimentally. The numerical study is conducted by means of potential theory, incorporating first and second order exciting wave forces, and their superposition to the calm water manoeuvring models. The applicability of such an approach is also investigated. The experimental work has been conducted at Flanders Hydraulics Research (in cooperation with Ghent University) with a scale model of an ultra large container vessel. Captive model tests comprise harmonic yaw tests and steady straight line tests with and without waves, at different forward speeds, wave frequencies and amplitudes, in head and following waves. Waves are chosen to represent conditions commonly met by ships in the Belgian coastal zone of the North Sea.

Influence of Wave Shape on the Impact of Shallow-Water Waves on Vertical Walls

2006 ◽  
Author(s):  
Claudio Zanzi ◽  
Pablo Go´mez ◽  
Julia´n Palacios ◽  
Joaqui´n Lo´pez ◽  
Julio Herna´ndez
Keyword(s):  
Shallow Water ◽  
Level Set ◽  
Water Waves ◽  
High Speed ◽  
Numerical Study ◽  
Local Level ◽  
Wave Shape ◽  
Shallow Water Waves ◽  
Vertical Walls ◽  
The Impact

A numerical study of the impact of shallow-water waves on vertical walls is presented. The air-liquid flow was simulated using a code for incompressible viscous flow, based on a local level set algorithm and a second-order approximate projection method. The level set transport and reinitialization equations were solved in a narrow band around the interface using an adaptive refined grid. The wave is assumed to be generated by a plunger which is accelerated in an open channel containing water. An arbitrary Lagrangian-Eulerian method was used to take into account the relative movement between the plunger and the end wall of the channel. The evolution of the free surface was visualized using a laser light sheet and a high-speed camera, with a sampling frequency of 1000 Hz. Several simulations were carried out to investigate the influence of the shape of the wave approaching the wall on the relevant quantities associated with the impact. The wave shape just before the impact was changed varying the total length of the channel. The results are compared with experimental results and with results obtained by other authors.

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