A conservative fourth-order stable finite difference scheme for the generalized Rosenau–KdV equation in both 1D and 2D

Dai, Weizhong

doi:10.1016/j.cam.2019.01.041

Cited by 23 publications

(11 citation statements)

References 28 publications

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“…Significant numerical studies have been done on the Rosenau-KdV equation [4,5]. Two-level nonlinear implicit Crank-Nicolson difference scheme and three-level linearimplicit difference scheme were presented to solve two-dimensional generalized Rosenau-KdV equation by Atouani [4].…”

Section: Introductionmentioning

confidence: 99%

“…Their experiment proved that both schemes were uniquely solvable, unconditionally stable and second-order convergent in L 1 norm, the linearized scheme was more effective in terms of accuracy and computational cost. Wang and Dai [5] proposed a conservative unconditionally stable finite difference scheme with O(h 4 +τ 2 ) for the generalized Rosenau-KdV equation in both one and two dimension, where h is spatial step and τ is temporal step, respectively.…”

Section: Introductionmentioning

confidence: 99%

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SSP IMEX Runge-Kutta WENO Scheme for Generalized Rosenau-KdV-RLW Equation

null¹,

Qiu²

2022

JMS

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In this article, we present a third-order weighted essentially non-oscillatory (WENO) method for generalized Rosenau-KdV-RLW equation. The third order finite difference WENO reconstruction and central finite differences are applied to discrete advection terms and other terms, respectively, in spatial discretization. In order to achieve the third order accuracy both in space and time, four stage third-order L-stable SSP Implicit-Explicit Runge-Kutta method (Third-order SSP EXRK method and thirdorder DIRK method) is applied to temporal discretization. The high order accuracy and essentially non-oscillatory property of finite difference WENO reconstruction are shown for solitary wave and shock wave for Rosenau-KdV and Rosenau-KdV-RLW equations. The efficiency, reliability and excellent SSP property of the numerical scheme are demonstrated by several numerical experiments with large CFL number.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

SSP IMEX Runge-Kutta WENO Scheme for Generalized Rosenau-KdV-RLW Equation

null¹,

Qiu²

2022

JMS

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“…Equation ( 3) is usually called the Rosenau-KdV equation. The authors of [17,18] proposed conservative schemes for the Rosenau-KdV equation based on the finite difference method. The authors of [19] proposed a Crank-Nicolson meshless spectral radial point interpolation (CN-MSRPI) method for the nonlinear Rosenau-KdV equation.…”

Section: Introductionmentioning

confidence: 99%

A Conservative and Implicit Second-Order Nonlinear Numerical Scheme for the Rosenau-KdV Equation

2021

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In this paper, for solving the nonlinear Rosenau-KdV equation, a conservative implicit two-level nonlinear scheme is proposed by a new numerical method named the multiple integral finite volume method. According to the order of the original differential equation’s highest derivative, we can confirm the number of integration steps, which is just called multiple integration. By multiple integration, a partial differential equation can be converted into a pure integral equation. This is very important because we can effectively avoid the large errors caused by directly approximating the derivative of the original differential equation using the finite difference method. We use the multiple integral finite volume method in the spatial direction and use finite difference in the time direction to construct the numerical scheme. The precision of this scheme is O(τ2+h3). In addition, we verify that the scheme possesses the conservative property on the original equation. The solvability, uniqueness, convergence, and unconditional stability of this scheme are also demonstrated. The numerical results show that this method can obtain highly accurate solutions. Further, the tendency of the numerical results is consistent with the tendency of the analytical results. This shows that the discrete scheme is effective.

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“…It is well known that the Korteweg-de Vries (KdV) equation describes the propagation of long waves on the surface of water with a small amplitude and is widely used to explain many complex science phenomena [1,2]. Various forms of the expansion for the KdV equation have been proposed because of its importance, such as the KdV-Burgers equation [3], the KdV-BBM equation [4], the Rosenau-KdV equation [5], the modified KdV equation [6], KdV-hierarchy [7], and the (2 + 1)-dimensional KdV equation [8]. In this research article, we consider the following (2 + 1)-dimensional integrable coupling of the KdV equation which has the bi-Hamiltonian structure for the (2 + 1)-dimensional perturbation equations of the KdV hierarchy [9]:…”

Section: Introductionmentioning

confidence: 99%

Construction of invariant solutions and conservation laws to the $(2+1)$-dimensional integrable coupling of the KdV equation

Gao

Yin

2020

Bound Value Probl

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Under investigation in this paper is the $(2+1)$ ( 2 + 1 ) -dimensional integrable coupling of the KdV equation which has applications in wave propagation on the surface of shallow water. Firstly, based on the Lie symmetry method, infinitesimal generators and an optimal system of the obtained symmetries are presented. At the same time, new analytical exact solutions are computed through the tanh method. In addition, based on Ibragimov’s approach, conservation laws are established. In the end, the objective figures of the solutions of the coupling of the KdV equation are performed.

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A conservative fourth-order stable finite difference scheme for the generalized Rosenau–KdV equation in both 1D and 2D

Cited by 23 publications

References 28 publications

SSP IMEX Runge-Kutta WENO Scheme for Generalized Rosenau-KdV-RLW Equation

SSP IMEX Runge-Kutta WENO Scheme for Generalized Rosenau-KdV-RLW Equation

A Conservative and Implicit Second-Order Nonlinear Numerical Scheme for the Rosenau-KdV Equation

Construction of invariant solutions and conservation laws to the $(2+1)$-dimensional integrable coupling of the KdV equation

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