Numerical Analysis of a Linear‐Implicit Average Scheme for Generalized Benjamin‐Bona‐Mahony‐Burgers Equation

Che, Haitao; Pan, Xintian; Zhang, Luming; Wang, Yiju

doi:10.1155/2012/308410

Cited by 11 publications

(6 citation statements)

References 19 publications

(1 reference statement)

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“…Bell-shaped and kink-shaped solutions of the generalized BBM-Burgers were found by the improved system technique [24]. Much effort has been devoted to the development of computational methods for the BBM-Burgers equation, including the finite difference methods (FDMs) [25][26][27][28][29][30], Adomian's decomposition methods [31], meshfree methods [32,33], collocation method [34], B-spline quasiinterpolation [35], and Galerkin cubic B-spline finite element method [36].…”

Section: Introductionmentioning

confidence: 99%

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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: Introductionmentioning

confidence: 99%

“…Due to the number of publications on numerical studies on both RLW and BBM-Burgers equations, much effort has been made to develop effective FDMs [17][18][19][25][26][27][28][29][30]. Most of the previous experiments, however, have the second order of precision in time and space.…”

Section: Introductionmentioning

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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“…Authors [2,3] studied the attenuation of its solution, and the existence, uniqueness and convergence of the solution of BBM equation are also proved in [4][5][6]. Various numerical methods also have attracted the attention of many researchers [7][8][9][10][11][12][13][14][15][16]. In this paper, the following initial value condition and boundary value conditions of BBM equation are considered.…”

Section: Introductionmentioning

confidence: 99%

A two-level high accuracy linearized difference scheme for the Benjamin-Bona-Mahony equation

Li-Jia

et al. 2022

THERM SCI

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“…Finite element Galerkin methods have been discussed by Kadri et al [24], Lee [25], Yin and Piao [26], and Dehghan et al [27] to solve the BBM-Burgers equations with a dissipative term. Finite difference methods have also been widely used to solve the equations in [28][29][30]; as for other numerical methods, refer to [31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

Finite Element Method of BBM-Burgers Equation with Dissipative Term Based on Adaptive Moving Mesh

Gao

et al. 2017

Discrete Dynamics in Nature and Society

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A finite element model is proposed for the Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation with a high-order dissipative term; the scheme is based on adaptive moving meshes. The model can be applied to the equations with spatial-time mixed derivatives and high-order derivative terms. In this scheme, new variables are needed to make the equation become a coupled system, and then the linear finite element method is used to discretize the spatial derivative and the fifth-order Radau IIA method is used to discretize the time derivative. The simulations of 1D and 2D BBM-Burgers equations with high-order dissipative terms are presented in numerical examples. The numerical results show that the method keeps a second-order convergence in space and provides a smaller error than that based on the fixed mesh, which demonstrates the effectiveness and feasibility of the finite element method based on the moving mesh. We also study the effect of the dissipative terms with different coefficients in the equation; by numerical simulations, we find that the dissipative term plays a more important role than in dissipation.

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Numerical Analysis of a Linear‐Implicit Average Scheme for Generalized Benjamin‐Bona‐Mahony‐Burgers Equation

Cited by 11 publications

References 19 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

A two-level high accuracy linearized difference scheme for the Benjamin-Bona-Mahony equation

Finite Element Method of BBM-Burgers Equation with Dissipative Term Based on Adaptive Moving Mesh

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