A least-squares finite element scheme for the RLW equation

Gardner, L. R. T.; Gardner, G. A.; Doğan, Abdülkadir

doi:10.1002/(sici)1099-0887(199611)12:11<795::aid-cnm22>3.0.co;2-o

Cited by 116 publications

(50 citation statements)

References 9 publications

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“…In Table 1 the time evolution of the invariants I 1 , I 2 and I 3 , and of the error norms L 2 and L 1 , is compared with previous works [8,9,[17][18][19]. It can be noticed that the proposed Petrov-Galerkin FEM formulation achieves superior conservation properties and accuracy: changes in I 1 at t ¼ 20 are below 10 À3 %, while I 2 and I 3 are exact up to the last recorded digit; the maximum error, L 1 -norm, is 0:026 Â 10 À3 while the L 2 -norm is 0:065 Â 10 À3 .…”

Section: Solitary Wave Propagationsupporting

confidence: 90%

“…For comparison with earlier results [8,9,[17][18][19], we take d ¼ l 2 ¼ 1. The numerical domain is x 2 0; 100 ½ with Dx ¼ 0:125 and Dt ¼ 0:1, and therefore Cr ¼ 0:8 and…”

Section: Solitary Wave Propagationmentioning

confidence: 95%

“…Let us now discretize equation (18) by means of the element shape functions (12) and of the weight functions (14), on a regular space-time finite element mesh (cf. Fig.…”

Section: Discrete Equationmentioning

confidence: 99%

“…Numerical curves for Cr ¼ 0:8 smaller than the same error for the least-squares linear FEM [18] and for the least-squares quadratic FEM [8].…”

Section: Solitary Wave Propagationmentioning

confidence: 99%

“…Several finite element schemes based on spline Galerkin techniques have been applied to the RLW equation, (see, e.g. [2,5,8,9,15,16,17,18,24]). Luo and Liu [21] proposed a mixed finite element formulation.…”

Section: Introductionmentioning

confidence: 99%

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A Petrov?Galerkin finite element scheme for the regularized long wave equation

Paulo

Seabra-Santos

2004

Computational Mechanics

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A Petrov-Galerkin finite element method (FEM) for the regularized long wave (RLW) equation is proposed. Finite elements are used in both the space and the time domains. Dispersion correction and a highly selective dissipation mechanism are introduced through additional streamline upwind terms in the weight functions. An implicit, conditionally stable, one-step predictor-corrector time integration scheme results. The accuracy and stability are investigated by means of local expansion by Taylor series and the resulting equivalent differential equation. An analysis based on a linear Fourier series solution and the Von Neumann's stability criterion is also performed. Based on the order of the analytical approximations and of the domain discretization it is concluded that the scheme is of third order in the nonlinear version and of fourth order in the linear version. Three numerical experiments of wave propagation are presented and their results compared with similar ones in the literature: solitary wave propagation, undular bore propagation, and cnoidal wave propagation. It is concluded that the present scheme possesses superior conservation and accuracy properties. IntroductionThe regularized long wave (RLW) equation was first proposed by Peregrine [23] for modelling the propagation of unidirectional weakly nonlinear and weakly dispersive water waves. Later on, Benjamin et al. [4] proposed the use of the RLW equation as a preferred alternative to the more classical Korteweg-de Vries (KdV) equation [20], to model a larger class of physical phenomena. These authors showed that the RLW equation is better posed than the KdV equation. Abdulloev et al.[1] numerically exposed the inelastic behaviour of solitary waves modelled by the RLW equation, which results from the fact that this model only possesses three conserved quantities, as demonstrated by Olver [22].Numerical solutions of the RLW equation based on the finite difference method were proposed by several authors (see, e.g. [13,14,19,23]). Spectral methods for the same equation were presented by Ben-Yu and Manoranjan [3] and Sloan [25]. Several finite element schemes based on spline Galerkin techniques have been applied to the RLW equation, (see, e.g. [2,5,8,9,15,16,17,18,24]). Luo and Liu [21] proposed a mixed finite element formulation. These finite element formulations were usually approximations with C 1 continuity. A finite volume method was introduced for a generalized KdV-RLW equation by Bradford and Sanders [7]. Finally, Durán and López-Marcos [11,12] showed the importance of conservative numerical methods for the long run simulation and the solitary wave interactions of the RLW model.Our purpose here is to formulate a Petrov-Galerkin finite element method based on a space-time finite element with C 0 continuity. The weight functions are derived from those proposed by Yu and Heinrich [26] for the convective-diffusion equation. It can be extended to bidimensional, multivariate systems [27] and thus amenable to be extended to the Boussinesq equations [6].The r...

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Section: Solitary Wave Propagationsupporting

confidence: 90%

“…For comparison with earlier results [8,9,[17][18][19], we take d ¼ l 2 ¼ 1. The numerical domain is x 2 0; 100 ½ with Dx ¼ 0:125 and Dt ¼ 0:1, and therefore Cr ¼ 0:8 and…”

Section: Solitary Wave Propagationmentioning

confidence: 95%

“…Let us now discretize equation (18) by means of the element shape functions (12) and of the weight functions (14), on a regular space-time finite element mesh (cf. Fig.…”

Section: Discrete Equationmentioning

confidence: 99%

“…Numerical curves for Cr ¼ 0:8 smaller than the same error for the least-squares linear FEM [18] and for the least-squares quadratic FEM [8].…”

Section: Solitary Wave Propagationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Petrov?Galerkin finite element scheme for the regularized long wave equation

Paulo

Seabra-Santos

2004

Computational Mechanics

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Simulations of the EW undular bore

Gardner

et al. 1997

Commun. Numer. Meth. Engng.

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SUMMARYThe equal width equation is solved by a Petrov±Galerkin method using quadratic B-spline spatial ®nite elements. A linear recurrence relationship for the numerical solution of the resulting system of ordinary dierential equations is obtained via a Crank±Nicolson approach involving a product approximation. The motion of solitary waves is studied to assess the properties of the algorithm. The development of an EW undular bore is investigated and compared with that of the RLW bore.

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A numerical solution of the RLW equation by Galerkin method using quartic B‐splines

Saka

Dağ

2007

Commun. Numer. Meth. Engng.

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SUMMARYGalerkin finite element method based on quartic B-splines is used to find a numerical solution of the regularized long wave equation. The method is tested on the problems of propagation of solitary waves, interaction of two and three solitary waves, undulation and wave generation. Comparisons are made with both analytical solutions and results of some published methods. Accuracy and efficiency of the scheme are discussed by computing numerical conserved laws and L 2 , L ∞ error norms.

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A least-squares finite element scheme for the RLW equation

Cited by 116 publications

References 9 publications

A Petrov?Galerkin finite element scheme for the regularized long wave equation

A Petrov?Galerkin finite element scheme for the regularized long wave equation

Simulations of the EW undular bore

A numerical solution of the RLW equation by Galerkin method using quartic B‐splines

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