Convergence of two conservative high-order accurate difference schemes for the generalized Rosenau–Kawahara-RLW equation

Ghiloufi, Ahlem; Rahmeni, Mohamed; Omrani, Khaled

doi:10.1007/s00366-019-00719-y

Cited by 22 publications

(5 citation statements)

References 27 publications

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“…Due to the method's preservation of physical properties, it had a similar or greater impact on numerical simulations [27,37,38,[41][42][43][44][45][46][47]. Therefore, in this example, we intended to provide and monitor the numerical simulations obtained in Examples 1 and 2 to validate the energy-preserving property discussed in Theorem 4.…”

Section: Conservative Approximationsmentioning

confidence: 99%

Analytical and numerical studies on dynamic of complexiton solution for complex nonlinear dispersive model

Phumichot,

Kuntiya,

Poochinapan

et al. 2023

Math Methods in App Sciences

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In this paper, we apply the basic one‐way propagator in order to introduce complex regularized long wave (RLW) equation. The complex RLW can be transformed into a system of nonlinear equations. By adopting a consequence nonlinear system, we can derive an energy conservation law of a complex regularized long wave equation. We then investigate how the Ansatz method may be applied to find a class of solitary wave solutions. Simultaneously, a numerical scheme for solving the model is implemented using a finite difference method based on the energy‐preserving Crank–Nicolson/Adams–Bashforth technique. It is worth mentioning that the obtained system is nonlinear. However, by using the present algorithm, we are able to linearize this system and solve it because of the implicit nature of the system of equations. An a priori estimate of the numerical solutions is derived to obtain a convergence and stability analysis; this yields second‐order accuracy in both time and space. Additionally, some numerical experiments verify computational efficiency. The results indicate that this method is an excellent way to preserve energy conservation, providing second‐order accuracy both in time and space with a maximum norm. In addition, we use the proposed scheme to study the effects of dispersive parameters when proceeding with an initial complex Gaussian condition.

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Section: Conservative Approximationsmentioning

confidence: 99%

Analytical and numerical studies on dynamic of complexiton solution for complex nonlinear dispersive model

Phumichot,

Kuntiya,

Poochinapan

et al. 2023

Math Methods in App Sciences

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“…In [32], He and Pan initially studied a second-order three-level linearly implicit difference scheme which is energy-conserved and unconditionally stable. In [33], two conservative high-order accurate finite difference schemes for the periodic initial value generalized Rosenau-Kawahara-RLW equation were introduced and extensively studied. For more related nonlinear wave equations, readers can refer to [34][35][36][37][38][39][40][41][42][43].…”

Section: Introductionmentioning

confidence: 99%

On Solitary Wave Solutions for the Camassa-Holm and the Rosenau-RLW-Kawahara Equations with the Dual-Power Law Nonlinearities

Sukantamala

Nanta

2021

Abstract and Applied Analysis

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The nonlinear wave equation is a significant concern to describe wave behavior and structures. Various mathematical models related to the wave phenomenon have been introduced and extensively being studied due to the complexity of wave behaviors. In the present work, a mathematical model to obtain the solution of the nonlinear wave by coupling the classical Camassa-Holm equation and the Rosenau-RLW-Kawahara equation with the dual term of nonlinearities is proposed. The solution properties are analytically derived. The new model still satisfies the fundamental energy conservative property as the original models. We then apply the energy method to prove the well-posedness of the model under the solitary wave hypothesis. Some categories of exact solitary wave solutions of the model are described by using the Ansatz method. In addition, we found that the dual term of nonlinearity is essential to obtain the class of analytic solution. Besides, we provide some graphical representations to illustrate the behavior of the traveling wave solutions.

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“…A high-order conservative difference scheme for nonlinear PDE's has been studied in [26][27][28][29][30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%

An efficient numerical simulation of nonlinear‐dispersive model of shallow water wave

2021

Self Cite

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This paper studied a linearized conservative high-order finite difference scheme for a model of nonlinear dispersive equation: the regularized long wave-Korteweg de Vries (RLW-KdV) equation. The scheme is proved to be conservative, uniquely solvable, and conditionally stable. The convergence of the difference scheme is proved by using the energy method to be of fourth-order in space and second-order in time in the discrete 𝐿 ∞ -norm. An application on the regularized long wave and the modified regularized long wave equations are discussed numerically in details. Furthermore, interaction of solitary waves with different amplitudes is shown. The three invariants of the motion are evaluated to determine the conservation proprieties of the system. Several numerical examples are presented in order to validate the theoretical results and compared with other existing methods. Comparison reveals that our linearized difference improves the accuracy and shortens computation time largely.

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Convergence of two conservative high-order accurate difference schemes for the generalized Rosenau–Kawahara-RLW equation

Cited by 22 publications

References 27 publications

Analytical and numerical studies on dynamic of complexiton solution for complex nonlinear dispersive model

Analytical and numerical studies on dynamic of complexiton solution for complex nonlinear dispersive model

On Solitary Wave Solutions for the Camassa-Holm and the Rosenau-RLW-Kawahara Equations with the Dual-Power Law Nonlinearities

An efficient numerical simulation of nonlinear‐dispersive model of shallow water wave

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