A one-parameter family of difference schemes for the regularized long-wave equation

Berikelashvili, Givi; Mirianashvili, Manana

doi:10.1515/gmj.2011.0044

Cited by 12 publications

(12 citation statements)

References 14 publications

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“…Note that the exact solution of the problem is

u (x, t) = 3 c s e c h^{2} (k_{0} [x - (c + 1) t - x_{0}]),

where c , x 0 are arbitrary constants and

k_{0} = \frac{1}{2} \sqrt{\frac{c}{μ false(1 + c false)}}

.We discretize the problem by the following FD scheme:

U_{t}^{n} - μ {boldH}_{1} U_{x \bar{x} t}^{n} + {boldH}_{2} U_{\hat{x}}^{n + \frac{1}{2}} + H_{2} Φ ({boldU}^{n + \frac{1}{2}}, {boldU}^{n + \frac{1}{2}}) = 0, n = 0, ⋯, N - 1,

U_{j}^{0} = u_{0} (x_{j}), j = 1, ⋯, J .

As it is shown in Berikelashvili and Mirianashvili, with the periodic boundary conditions for a region a ≤ x ≤ b , the RLW equation has 3 conservation laws, which can be expressed in the following form

I_{1} = \int_{a}^{b} u d x,

I_{2} = \int_{a}^{b} ((), u^{2} + μ u_{x}^{2})

…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…We consider, in this case, the RLW equation with initial condition given by the linear sum of 2 well‐separated solitary waves of various amplitudes

u false(x, 0 false) = true \sum_{i = 1}^{2} 3 c_{i} s e c h^{2} ((), k_{i} false(x - x_{i} false)),

where

c_{i} = \frac{4 k_{i}^{2}}{1 - 4 k_{i}^{2}}, c_{i}

and x i are constants, i = 1,2(see Berikelashvili and Mirianashvili). In this section, we choose μ = 1, k 1 = 0.4, k 2 = 0.3, x 1 = 15, x 2 = 35, h = 0.3,and k = 0.1 with interval [0,120].…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…where c, x 0 are arbitrary constants and k 0 = 1 2 √ c (1+c) .We discretize the problem (72)-(73) by the following FD scheme: As it is shown in Berikelashvili and Mirianashvili, 14 with the periodic boundary conditions for a region a ≤ x ≤ b, the RLW equation has 3 conservation laws, which can be expressed in the following form…”

Section: Example 3: Application On the Rlw Equationmentioning

confidence: 99%

“…As we can see from the table, our scheme gives quite similar values to those enlisted in Raslan 40 and Kutluay and Esen. 13 The interactions of these 2 solitary waves are plotted at different time levels and the phenomenon of collision of solitons, which is discussed in Berikelashvili and Mirianashvili, 14 can be observed in Figure 2.…”

Section: Interaction Of 2 Solitary Wavesmentioning

confidence: 99%

“…An analytical solution for the RLW equation was found under the restricted initial and boundary conditions in Benjamin et al Since the analytical solution of the RLW equation is not very useful, the availability of accurate and efficient numerical methods are needed. In the recent years, several numerical methods for the solution of the RLW equation have been developed, including the finite difference (FD) method, the Fourier pseudospectral method, and the finite element methods based on Galerkin and collocation techniques . Achouri, Khiari, and Omrani constructed and studied some FD schemes for the RLW equation.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A new conservative fourth‐order accurate difference scheme for solving a model of nonlinear dispersive equations

Ghiloufi

Rouatbi

Omrani

2018

Math Methods in App Sciences

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This article is devoted to the study of a nonlinear conservative fourth‐order difference scheme for a model of nonlinear dispersive equations that is governed by the RLW‐KdV equation. The existence of the approximate solution and the convergence of the difference scheme are proved, by using the energy method. In addition, the convergent order in maximum norm is 2 in temporal direction and 4 in spatial direction. The unconditional stability as well as uniqueness of the difference scheme is also derived. An application on the RLW and MRLW equations is discussed numerically in details. Furthermore, interaction of solitary waves with different amplitudes are shown. The 3 invariants of the motion are evaluated to determine the conservation proprieties of the system. The temporal evaluation of a Maxwellian initial pulse is then studied. Some numerical examples are given to validate the theoretical results.

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“…Note that the exact solution of the problem is

u (x, t) = 3 c s e c h^{2} (k_{0} [x - (c + 1) t - x_{0}]),

where c , x 0 are arbitrary constants and

k_{0} = \frac{1}{2} \sqrt{\frac{c}{μ false(1 + c false)}}

.We discretize the problem by the following FD scheme:

U_{t}^{n} - μ {boldH}_{1} U_{x \bar{x} t}^{n} + {boldH}_{2} U_{\hat{x}}^{n + \frac{1}{2}} + H_{2} Φ ({boldU}^{n + \frac{1}{2}}, {boldU}^{n + \frac{1}{2}}) = 0, n = 0, ⋯, N - 1,

U_{j}^{0} = u_{0} (x_{j}), j = 1, ⋯, J .

I_{1} = \int_{a}^{b} u d x,

I_{2} = \int_{a}^{b} ((), u^{2} + μ u_{x}^{2})

…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…We consider, in this case, the RLW equation with initial condition given by the linear sum of 2 well‐separated solitary waves of various amplitudes

u false(x, 0 false) = true \sum_{i = 1}^{2} 3 c_{i} s e c h^{2} ((), k_{i} false(x - x_{i} false)),

where

c_{i} = \frac{4 k_{i}^{2}}{1 - 4 k_{i}^{2}}, c_{i}

and x i are constants, i = 1,2(see Berikelashvili and Mirianashvili). In this section, we choose μ = 1, k 1 = 0.4, k 2 = 0.3, x 1 = 15, x 2 = 35, h = 0.3,and k = 0.1 with interval [0,120].…”

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Example 3: Application On the Rlw Equationmentioning

confidence: 99%

Section: Interaction Of 2 Solitary Wavesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A new conservative fourth‐order accurate difference scheme for solving a model of nonlinear dispersive equations

Ghiloufi

Rouatbi

Omrani

2018

Math Methods in App Sciences

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On the convergence of difference schemes for generalized Benjamin–Bona–Mahony equation

Berikelashvili

Mirianashvili²

2013

Numerical Methods Partial

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We consider an initial boundary‐value problem for the generalized Benjamin–Bona–Mahony equation. A three‐level conservative difference schemes are studied. The obtained algebraic equations are linear with respect to the values of unknown function for each new level. It is proved that the scheme is convergent with the convergence rate of order k – 1, when the exact solution belongs to the Sobolev space of order k, (1

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The numerical analysis of two linearized difference schemes for the Benjamin–Bona–Mahony–Burgers equation

Zhang

Liu

Zhang

2020

Numerical Methods Partial

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In the article, two linearized finite difference schemes are proposed and analyzed for the Benjamin-Bona-Mahony-Burgers (BBMB) equation. For the construction of the two-level scheme, the nonlinear term is linearized via averaging k and k + 1 floor, we prove unique solvability and convergence of numerical solutions in detail with the convergence order O(2 + h 2). For the three-level linearized scheme, the extrapolation technique is utilized to linearize the nonlinear term based on function. We obtain the conservation, boundedness, unique solvability and convergence of numerical solutions with the convergence order O(2 + h 2) at length. Furthermore, extending our work to the BBMB equation with the nonlinear source term is considered and a Newton linearized method is inserted to deal with it. The applicability and accuracy of both schemes are demonstrated by numerical experiments.

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A one-parameter family of difference schemes for the regularized long-wave equation

Cited by 12 publications

References 14 publications

A new conservative fourth‐order accurate difference scheme for solving a model of nonlinear dispersive equations

A new conservative fourth‐order accurate difference scheme for solving a model of nonlinear dispersive equations

On the convergence of difference schemes for generalized Benjamin–Bona–Mahony equation

The numerical analysis of two linearized difference schemes for the Benjamin–Bona–Mahony–Burgers equation

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