Conservation laws and invariants of motion for nonlinear internal waves: part II

Hamdi, Samir; Morse, Brian; Halphen, Bernard; Schiesser, William E.

doi:10.1007/s11069-011-9737-4

Cited by 16 publications

(7 citation statements)

References 23 publications

(19 reference statements)

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“…To be able to make comparison between numerical and analytical solutions, solitary wave solution of GE is studied to see advance of wave during running time of the algorithm, which is defined analytically in some studies [1, 10] as

\begin{array}{c} u (x, t) goodbreak= S \sec h ((), k ((), x goodbreak- x_{0} goodbreak- italicct)) \\ k goodbreak= \sqrt{\frac{c}{μ_{3}}}, S goodbreak= \frac{6 c}{μ_{1} ((), 1 goodbreak+ \sqrt{1 + 6 \frac{μ_{2} c}{μ_{1}^{2}}})} . \end{array}

With parameters

μ_{1} = 1,

μ_{2} = 1

μ_{3} = 5

, and

x_{0} = 0,

initial profiles, having bell‐type shape is locate along the

x

‐axis centered at

x = 0

in the restricted region

false[- 100, 100 false]

. Three BCs are used to have solvable system of algebraic equations: two of them are on left,

u false(- 100, t false)

and

u_{x} false(- 100, t false)

and one is on the right,

u false(100, t false) = 0

.…”

Section: Numerical Examplesmentioning

confidence: 99%

A quartic trigonometric tension B‐spline finite element method for solving Gardner equation

Hepson

2022

Numerical Methods Partial

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The collocation method is established for getting solution of the Gardner equation (GE). The GE is fully integrated by way of the Crank-Nicolson method for time variable and collocation method for spatial variable. Trial function of the collocation method is set up using combination of quartic trigonometric tension (QTT) B-splines. Convergence analysis of suggested method is investigated. Performance of the QTT B-splines is searched by studying three test problems: propagation of bell shape solitary wave, interaction of two positive bell shape solitary wave and wave generation.

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\begin{array}{c} u (x, t) goodbreak= S \sec h ((), k ((), x goodbreak- x_{0} goodbreak- italicct)) \\ k goodbreak= \sqrt{\frac{c}{μ_{3}}}, S goodbreak= \frac{6 c}{μ_{1} ((), 1 goodbreak+ \sqrt{1 + 6 \frac{μ_{2} c}{μ_{1}^{2}}})} . \end{array}

With parameters

μ_{1} = 1,

μ_{2} = 1

μ_{3} = 5

, and

x_{0} = 0,

initial profiles, having bell‐type shape is locate along the

x

‐axis centered at

x = 0

in the restricted region

false[- 100, 100 false]

. Three BCs are used to have solvable system of algebraic equations: two of them are on left,

u false(- 100, t false)

and

u_{x} false(- 100, t false)

and one is on the right,

u false(100, t false) = 0

.…”

Section: Numerical Examplesmentioning

confidence: 99%

A quartic trigonometric tension B‐spline finite element method for solving Gardner equation

Hepson

2022

Numerical Methods Partial

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“…are expected to keep their initial values as time proceeds [19]. In order to measure the absolute relative changes of these quantities at any time t > 0, C(M t ), C(E t ) and C(H t ) are defined as…”

Section: Illustrationsmentioning

confidence: 99%

“…The three conservation laws representing various physical quantities such as momentum, energy and etc. are given for the generalized Gardner equation power law nonlinearities by using some algebraic and derivative manipulations [19]. Some finite difference and restrictive Taylor's approximation are used to determine the numerical solutions to the Gardner equation [20,21].…”

Section: Introductionmentioning

confidence: 99%

Exponential B-spline collocation solutions to the Gardner equation

Hepson

Korkmaz

Dağ

2019

International Journal of Computer Mathematics

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The exponential B-spline basis function set is used to develop a collocation method for some initial boundary value problems (IBVPs) to the Gardner equation. The Gardner equation has two nonlinear terms, namely quadratic and cubic ones. The order reduction of the equation is resulted in a coupled system of PDEs that enables the exponential B-splines to be implemented. The system is integrated in time by Crank-Nicolson implicit method. The validity of the method is investigated by calculating the discrete maximum error norm and observing the absolute relative changes of the conservation laws at the end of the simulations.

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“…are expected to keep their initial quantities during numerical simulations [22]. The relative changes of these quantities at a discrete time t > 0 are measured by using C(M t ), C(E t ) and C(H t ) defined as…”

Section: Numerical Illustrationsmentioning

confidence: 99%

Numerical solutions of the Gardner equation by extended form of the cubic B-splines

2018

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The extended definition of the polynomial B-splines may give a chance to improve the results obtained by the classical cubic polynomial B-splines. Determination of the optimum value of the extension parameter can be achieved by scanning some intervals containing zero. This study aims to solve some initial boundary value problems constructed for the Gardner equation by the extended cubic B-spline collocation method. The test problems are derived from some analytical studies to validate the efficiency and accuracy of the suggested method. The conservation laws are also determined to observe them remain constant as expected in theoretical aspect. The stability of the proposed method is investigated by the Von Neumann analysis.

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Conservation laws and invariants of motion for nonlinear internal waves: part II

Cited by 16 publications

References 23 publications

A quartic trigonometric tension B‐spline finite element method for solving Gardner equation

A quartic trigonometric tension B‐spline finite element method for solving Gardner equation

Exponential B-spline collocation solutions to the Gardner equation

Numerical solutions of the Gardner equation by extended form of the cubic B-splines

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