New solitary solutions and a conservative numerical method for the Rosenau–Kawahara equation with power law nonlinearity

He, Dongdong

doi:10.1007/s11071-015-2224-9

Cited by 25 publications

(4 citation statements)

References 62 publications

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“…Remark 16 In Schemes II and III the difference approximation of the nonlinear term uu x is widely used to ensure the energy conservation, stability, and convergence of the numerical solutions. For further information, the reader should consult references [43][44][45][46][47] for various types of equations with the same treatments of the nonlinear term uu x .…”

Section: Numerical Experimentsmentioning

confidence: 99%

Novel advances in high-order numerical algorithm for evaluation of the shallow water wave equations

Poochinapan

Wongsaijai

2023

Adv Cont Discr Mod

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In this paper, we propose a high-order nonlinear algorithm based on a finite difference method modification to the regularized long wave equation and the Benjamin–Bona–Mahony–Burgers equation subject to the homogeneous boundary. The consequence system of nonlinear equations typically trades with high computation burden. This dilemma can be overcome by establishing a fast numerical algorithm procedure without a reduction of numerical accuracy. The proposed algorithm forms a linear system with constant coefficient matrix at each time step and produces numerical solutions, which remarkably gains many computational advantages. In terms of analysis, a priori estimation for the numerical solution is derived to obtain the convergence and stability analysis. Additionally, the algorithm is globally mass preserving to avoid nonphysical behavior. Two benchmarks, including a single solitary wave to both equations, are given to validate the applicability and accuracy of the proposed method. Numerical results are obtained and compared to other approaches available in the literature. From the comparisons it is clear that the proposed approach produces accurate and precise results at low computational cost. Besides, the proposed scheme is applied to study the effect of the viscous term on a single solitary wave. It is shown that the viscous term results in the amplitude and width of the initial condition but not in its velocities in the case of a single solitary wave. As a consequence, theoretical and numerical findings provide a new area to investigate and expand the high-order algorithm for the family of wave equations.

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Section: Numerical Experimentsmentioning

confidence: 99%

Novel advances in high-order numerical algorithm for evaluation of the shallow water wave equations

Poochinapan

Wongsaijai

2023

Adv Cont Discr Mod

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“…There are numerical observations supporting the generalized Rosenau-Kawahara equation with power law nonlinearity. 26,27 He 27 furnished a set of exact solitary solutions and a three-level second-order conservative finite difference technique. Besides, using a simplified three-level implicit linear conservative finite difference scheme and a two-level nonlinear Crank-Nicolson difference scheme to solve the usual Rosenau-Kawahara equation of Hu et al 28 succeeded in showing that two conservative finite difference schemes are of second-order convergence unconditionally stable.…”

Section: Introductionmentioning

confidence: 99%

Advance in compact structure‐preserving manner to the Rosenau–Kawahara model of shallow‐water wave

Wongsaijai

Charoensawan

Chaobankoh

et al. 2021

Math Methods in App Sciences

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In this paper, we pursue further analysis of the performance of a compact structure-preserving finite difference scheme. The convergence, stability, and accuracy of the approximate solution with respect to grid refinement are discussed. The compact difference approach to precisely preserving invariants on any time-space regions gives a three-level linear-implicit scheme with the spatial accuracy, found to be fourth order on a uniform grid. The method is verified by comparison with a solution of the Rosenau-Kawahara equation just obtained with second-order finite difference schemes recently. Also, the efficiency of the present algorithm is confirmed by simulations of the problem at a long time. Details of CPU time are examined in order to assess the usefulness of the compact scheme for determining an approximate solution.

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“…The dynamics of dispersive shallow water waves is extensively studied by various known models. These are the Rosenau Kawahara equation , Rosenau‐KdV equation and Rosenau‐RLW equation and many others.…”

Section: Introductionmentioning

confidence: 99%

New conservative difference schemes with fourth‐order accuracy for some model equation for nonlinear dispersive waves

Ghiloufi

Omrani

2017

Numerical Methods Partial

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In this article, some high‐order accurate difference schemes of dispersive shallow water waves with Rosenau‐KdV‐RLW‐equation are presented. The corresponding conservative quantities are discussed. Existence of the numerical solution has been shown. A priori estimates, convergence, uniqueness, and stability of the difference schemes are proved. The convergence order is O ( h 4 + k 2 ) in the uniform norm without any restrictions on the mesh sizes. At last numerical results are given to support the theoretical analysis.

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New solitary solutions and a conservative numerical method for the Rosenau–Kawahara equation with power law nonlinearity

Cited by 25 publications

References 62 publications

Novel advances in high-order numerical algorithm for evaluation of the shallow water wave equations

Novel advances in high-order numerical algorithm for evaluation of the shallow water wave equations

Advance in compact structure‐preserving manner to the Rosenau–Kawahara model of shallow‐water wave

New conservative difference schemes with fourth‐order accuracy for some model equation for nonlinear dispersive waves

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