Numerical solution of regularized long wave equation using Petrov–Galerkin method

Doğan, Abdülkadir

doi:10.1002/cnm.424

Cited by 33 publications

(22 citation statements)

References 5 publications

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“…We find that higher accuracy is obtained than the results in the paper [7, Table IE] when smaller space-time steps were used. For instance, in paper [7], when h = 0.025 and Af = 0.025, error norms L 2 = 19.9 x 10~3 and L x = 5.87 x 10" 3 were obtained at time t = 40 whereas we get L 2 = 1.110 x 10" 3 and L x = 0.431 x 10~3. Secondly, we study the undular bore development using the initial condition Table 4.…”

Section: J-oo J-oo J-oomentioning

confidence: 83%

“…Finite difference methods were proposed based on both cubic and quintic splines in the papers [3,11]. Some variants of finite element and collocation methods were constructed for the RLW equation by using B-splines as weight and trial functions [6][7][8][9]. The use of cubic splines in numerical methods for finding solutions of PDEs leads to a matrix system which is tridiagonal, thus permitting the use of the Thomas algorithm.…”

Section: U X (At) = 0 U X (Bt) = 0 ---And Initial Conditionsmentioning

confidence: 99%

“…Firstly we will study the of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446181100009822 [7] Table 1. During the run, numerical invariants remained almost the same when compared with the analytical values (3.4).…”

Section: J-oo J-oo J-oomentioning

confidence: 99%

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Numerical integration of the RLW equation using cubic splines

2005

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Section: J-oo J-oo J-oomentioning

confidence: 83%

Section: U X (At) = 0 U X (Bt) = 0 ---And Initial Conditionsmentioning

confidence: 99%

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Numerical integration of the RLW equation using cubic splines

2005

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“…and the boundary conditions To compare our scheme with the recent methods [7,9,[12][13][14][16][17][18][19][20][21], the spatial grid spacing x = 0.125 and time step t = 0.1 are taken. Errors in L 2 and L ∞ norms and the three invariants C 1 , C 2 , C 3 for the two cases taken at various times are given in Tables I and II, [13,14,[16][17][18][19][20] and in the order of 10 −5 based on the pseudo-spectral method [9].…”

Section: Propagation Of a Solitary Wavementioning

confidence: 99%

“…This problem has been extensively studied by many authors [7,9,[12][13][14][16][17][18][19][20][21].…”

Section: Propagation Of a Solitary Wavementioning

confidence: 99%

High‐order compact difference scheme for the regularized long wave equation

Lin

Xie

Zhou

2006

Commun. Numer. Meth. Engng.

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SUMMARYA numerical simulation of the regularized long wave (RLW) equation is obtained using a high-order compact difference method, based on the fourth-order compact difference scheme in space and the fourthorder Runge-Kutta method in time integration. The method is tested on the problems of propagation of a solitary wave, interaction of two positive solitary waves, interaction of a positive and a negative solitary wave, the evaluation of Maxwellian pulse into stable solitary waves, the development of an undular bore and the solitary waves induced by boundary motion. The three invariants of the motion are calculated to determine the conservation properties of the algorithm. L 2 and L ∞ error norms are used to measure differences between the exact and numerical solutions. The results obtained by proposed method are compared with those of other recently published methods and shown to be more accurate and efficient.

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A linearized implicit pseudo‐spectral method for some model equations: the regularized long wave equations

Djidjeli

Price

Twizell

et al. 2003

Commun. Numer. Meth. Engng.

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SUMMARYAn e cient numerical method is developed for the numerical solution of non-linear wave equations typiÿed by the regularized long wave equation (RLW) and its generalization (GRLW). The method developed uses a pseudo-spectral (Fourier transform) treatment of the space dependence together with a linearized implicit scheme in time.An important advantage to be gained from the use of this method, is the ability to vary the mesh length, thereby reducing the computational time. Using a linearized stability analysis, it is shown that the proposed method is unconditionally stable. The method is second order in time and all-order in space.The method presented here is for the RLW equation and its generalized form, but it can be implemented to a broad class of non-linear long wave equations (Equation (2)), with obvious changes in the various formulae.Test problems, including the simulation of a single soliton and interaction of solitary waves, are used to validate the method, which is found to be accurate and e cient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm.

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Numerical solution of regularized long wave equation using Petrov–Galerkin method

Cited by 33 publications

References 5 publications

Numerical integration of the RLW equation using cubic splines

Numerical integration of the RLW equation using cubic splines

High‐order compact difference scheme for the regularized long wave equation

A linearized implicit pseudo‐spectral method for some model equations: the regularized long wave equations

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