Conservative Difference Scheme for Generalized Rosenau-KdV Equation

Luo, Yiyong; Xu, Yongbin; Feng, Meiqiang

doi:10.1155/2014/986098

Cited by 10 publications

(14 citation statements)

References 20 publications

(25 reference statements)

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“…All computations have been done using MATLAB R2011a on Intel (R) Core(TM) i7 CPU Q 720@1.60Ghz machine with 4 GB of memory. The initial boundary value problem given by Equations (1)-(3) has the following invariants [6,14]…”

Section: Numerical Examples and Resultsmentioning

confidence: 99%

“…In order to see how well the discrete invariants Q(t) and E(t) for the Lie-Trotter splitting and the Strang splitting techniques of Example 2 at various values of h = Δt at t = 0, 20, 40 are preserved, a comparison with those given in Ref. [6] is presented in Table 9. It is clearly seen from the table that both schemes preserve the invariants very well, and they are also in very good agreement with those given in Ref.…”

Section: Numerical Examples and Resultsmentioning

confidence: 99%

“…Razborova et al [5] studied dynamical features of dispersive shallow water waves modelled by the Rosenau-KdV-RLW equation. Luo et al [6] presented a conservative Crank-Nicolson finite difference scheme for the initial-boundary value problem of the generalized Rosenau-KdV equation and observed that the difference scheme shows a discrete analog of the main conservation laws associated to the equation. Zheng and Zhou [7] proposed an average linear finite difference scheme for the numerical solution of the initial-boundary value problem of the generalized Rosenau-KdV equation.…”

Section: Introductionmentioning

confidence: 99%

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Operator time‐splitting techniques combined with quintic B‐spline collocation method for the generalized Rosenau–KdV equation

Kutluay

Karta

Yağmurlu

2019

Numerical Methods Partial

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In this article, the generalized Rosenau-KdV equation is split into two subequations such that one is linear and the other is nonlinear. The resulting subequations with the prescribed initial and boundary conditions are numerically solved by the first order Lie-Trotter and the second-order Strang time-splitting techniques combined with the quintic B-spline collocation by the help of the fourth order Runge-Kutta (RK-4) method. To show the accuracy and reliability of the proposed techniques, two test problems having exact solutions are considered. The computed error norms L 2 and L ∞ with the conservative properties of the discrete mass Q(t) and energy E(t) are compared with those available in the literature. The convergence orders of both techniques have also been calculated. Moreover, the stability analyses of the numerical schemes are investigated.

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Section: Numerical Examples and Resultsmentioning

confidence: 99%

Section: Numerical Examples and Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Operator time‐splitting techniques combined with quintic B‐spline collocation method for the generalized Rosenau–KdV equation

Kutluay

Karta

Yağmurlu

2019

Numerical Methods Partial

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“…Therefore, we get ( e n+1 2 − e n 2 ) + ( e n+1 xx 2 − e n xx 2 ) = τ r n , 2e According to Theorem 9 in reference [7], Theorem 2.7 in reference [18], and Cauchy-Schwartz inequality, we have…”

Section: Lemma 32 (Discrete Sobolev Inequality)mentioning

confidence: 92%

“…In [7,16], two conservative difference schemes for the generalized Rosenau-KdV equation were proposed, while both only discussed one conservative law. Another conservative Crank-Nicolson implicit difference scheme was presented in [18], but the shortcoming exists in the computation for the initial condition, which needs the help of other scheme, such as the average linear scheme (see, for example, [16]).…”

Section: Introductionmentioning

confidence: 99%

Study on convergence and stability of a conservative difference scheme for the generalized Rosenau-KdV equation

Jun¹,

Zheng²,

Dai³

2017

J. Nonlinear Sci. Appl.

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In this paper, a conservative nonlinear implicit finite difference scheme for the generalized Rosenau-KdV equation is studied. Convergence and stability of the proposed scheme are proved by a discrete energy method. The proof with a priori error estimate shows that the convergence rates of numerical solutions are both the second order on time and in space. Meanwhile, numerical experiments are carried out to verify the theoretical analysis and show that the scheme is efficient and reliable.

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Numerical solutions of generalized Rosenau–KDV–RLW equation by using Haar wavelet collocation approach coupled with nonstandard finite difference scheme and quasilinearization

Verma

Rawani

2022

Numerical Methods Partial

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In this article, we analyze and propose to compute the numerical solutions of a generalized Rosenau-KDV-RLW (Rosenau-Korteweg De Vries-Regularized Long Wave) equation based on the Haar wavelet (HW) collocation approach coupled with nonstandard finite difference (NSFD) scheme and quasilinearization. In the process of the numerical solution, the NSFD scheme is applied to discretize the first-order time derivative, Haar wavelets are applied on spatial derivatives and the non-linear term is taken care by quasilinearization technique. To discuss the efficiency of the method we compute L ∞ error and L 2 error. We also use discrete mass and energy conservation to check the accuracy of the proposed methodology. The computed results have been compared with the existing methods, for example, three-level average implicit finite difference technique, B-spline collocation, three-level linear conservative implicit finite difference scheme and conservative fourth-order stable finite difference scheme.

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Conservative Difference Scheme for Generalized Rosenau-KdV Equation

Cited by 10 publications

References 20 publications

Operator time‐splitting techniques combined with quintic B‐spline collocation method for the generalized Rosenau–KdV equation

Operator time‐splitting techniques combined with quintic B‐spline collocation method for the generalized Rosenau–KdV equation

Study on convergence and stability of a conservative difference scheme for the generalized Rosenau-KdV equation

Numerical solutions of generalized Rosenau–KDV–RLW equation by using Haar wavelet collocation approach coupled with nonstandard finite difference scheme and quasilinearization

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