Simulations of solitons using quadratic spline finite elements

Gardner, L. R. T.; Gardner, G. A.; Ali, A.H.A.

doi:10.1016/0045-7825(91)90241-w

Cited by 33 publications

(25 citation statements)

References 10 publications

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“…= 0.392 8, where a denotes the amplitudes. When the parameter µ = 0.01 is used with t = 0.01, h = 0.1 and −15 ≤ x ≤ 15, three solutions are formed at time t = 10, which agrees with earlier work [31], Fig. 9.…”

Section: Examplesupporting

confidence: 69%

“…We solve five test examples to illustrate the efficiency of the proposed methods by studying the motion of solitons, interaction of two separated solitons, and soliton generation from any arbitrary initial condition. [3,5,7,[9][10][11]27]. To test the efficiency of the numerical scheme, we use the L 2 , L ∞ errors norms given in the Eqs.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Arnold and Winther [6] have introduced a superconvergent finite element method for the solutions of the KdVE. Finite difference schemes [7], B-spline finite element schemes [8,9], collocation method based on radial basis functions [10], and septic spline collocation method [11] are described to obtain numerical solutions. Shen [12] has simulated the motion of single soliton, interaction of five solitary waves, and wave generation from initial pulse solutions of the KdVE, assuming µ = 1, with high accuracy via Dual-Petrov-Galerkin method.…”

Section: Introductionmentioning

confidence: 99%

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Numerical algorithms for solutions of Korteweg–de Vries equation

Korkmaz¹

2009

Numerical Methods Partial

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The nonlinear Korteweg-de Vries (KdVE) equation is solved numerically using both Lagrange polynomials based differential quadrature and cosine expansion-based differential quadrature methods. The first test example is travelling single solitary wave solution of KdVE and the second test example is interaction of two solitary waves, whereas the other three examples are wave production from solitary waves. Maximum error norm and root mean square error norm are computed, and numerical comparison with some earlier works is done for the first two examples, the lowest four conserved quantities are computed for all test examples.

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Section: Examplesupporting

confidence: 69%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical algorithms for solutions of Korteweg–de Vries equation

Korkmaz¹

2009

Numerical Methods Partial

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“…The function Y ( x ) then varies over the element as (7) where N , ( ( ) = 4(1 -5) and N q ( t ) = 1 -r2…”

Section: Co-ordinate Transformationmentioning

confidence: 99%

Hermite infinite elements and graded quadratic B‐spline finite elements

Gardner

Dağ

1993

Numerical Meth Engineering

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SUMMARYQuadratic B-spline finite elements are defined for a graded mesh. Hermite infinite elements are proposed to dxtend the applicability of these finite elements to unbounded regions. Test problems used to compare this dechnique with published procedures show that the quadratic B-spline finite element solution has, as expected, lower error bounds than a linear element solution. These experiments also demonstrate that the Hermite infinite elements used to close the B-spline finite element arrays lead to error norms comparable in size with other infinite element formulations.The generation of solitary waves in a semi-infinite shallow channel by boundary forcing is modelled by the Korteweggde Vries equation using an array of graded elements closed by a zero pole infinite element. The resulting simulation of solitary wave motion across a non-uniform mesh confirms existing work and illustrates the effectiveness of the present formulation.

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“…For the fourth test problem we have selected the Maxwellian initial condition given by [18] U (x, 0) = exp…”

Section: Maxwellian Initial Conditionmentioning

confidence: 99%