Convergence and Stability in Maximum Norms of Linearized Fourth-Order Conservative Compact Scheme for Benjamin–Bona–Mahony–Burgers’ Equation

Zhang, Qifeng; Liu, Lingling

doi:10.1007/s10915-021-01474-3

Cited by 19 publications

(3 citation statements)

References 49 publications

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“…It is worth to note that, by using the Taylor expansion, it can be seen from scheme ( 15)-( 16) to compute U 1 and V 1 has the second-order accuracy in space but first-order in time if the function to be approximated is sufficiently smooth. However, with the help of the arguments in Zhang et al [35,36], we can prove that the global convergence order for scheme ( 11)-( 14) does not affect by the terms U 1 and V 1 (see Theorem 5).…”

Section: Notationsmentioning

confidence: 85%

“…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. Consequently, we turn our attention to the discrete energy produced by our present scheme as defined by Equation (36). Variations of discrete energy E n , and the residuals of conservative invariant E n as time evolved during the present scheme for m = 2 and m = 4 using h = 𝜏 = 0.05 and h = 𝜏 = 0.0125, are reported in Tables 3 and 4, respectively.…”

Section: Conservative Approximationsmentioning

confidence: 99%

See 1 more Smart Citation

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: Notationsmentioning

confidence: 85%

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

View full text Add to dashboard Cite

show abstract

“…Due to the nonlinearity of BBMB equation, it is very difficult to find out the true solution. Thus, a lot of numerical methods have been considered, such as the finite difference methods [8][9][10][11], collocation method [12], meshless method [13,14], finite element method (FEM) [15][16][17][18], and so on. For the FEM, Kadri [15] proposed semi-discrete and two kinds of fully discrete Galerkin schemes, studied the L ∞ -norm error estimates; Kundu [16] established the convergence of unsteady solution to steady state solution; Karakoc [17] obtained the convergence analysis by use of a cubic B-spline FEM; Gao [18] discussed the local discontinuous Galerkin FEM and derived an optimal error estimate.…”

Section: Introductionmentioning

confidence: 99%

Unconditional superconvergence analysis of an energy-stable finite element scheme for nonlinear Benjamin–Bona–Mahony–Burgers equation

2022

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In this paper, an energy-stable Crank–Nicolson fully discrete finite element scheme is proposed for the Benjamin–Bona–Mahony–Burgers equation. Firstly, the stability of energy is proved, which leads to the boundedness of the finite element solution in $H^{1}$ H 1 -norm. Secondly, combining with the above boundedness and the special property of bilinear element, the unconditional superclose and superconvergence results are derived. Finally, numerical examples are provided to illustrate the validity and efficiency of our theoretical analysis and method.

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A New Highly Accurate Numerical Scheme for Benjamin–Bona–Mahony–Burgers Equation Describing Small Amplitude Long Wave Propagation

2023

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Convergence and Stability in Maximum Norms of Linearized Fourth-Order Conservative Compact Scheme for Benjamin–Bona–Mahony–Burgers’ Equation

Cited by 19 publications

References 49 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

Unconditional superconvergence analysis of an energy-stable finite element scheme for nonlinear Benjamin–Bona–Mahony–Burgers equation

A New Highly Accurate Numerical Scheme for Benjamin–Bona–Mahony–Burgers Equation Describing Small Amplitude Long Wave Propagation

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