On the convergence of difference schemes for the generalized BBM–Burgers equation

Berikelashvili, Givi; Mirianashvili, Manana

doi:10.1515/gmj-2018-0075

Cited by 3 publications

(3 citation statements)

References 10 publications

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“…As we see from Figure 1a,b, the wave peak of solutions of the BBMB equation is dissipative when the time elapses. Similar results are observed by (6.1)–(6.4) in [42] and (4.10)–(4.13), which further confirm our schemes are efficient.Example We consider (5.1)–(5.3) with F ( u ) = 1/4 · (1 − u 2 ) 2 , x l = − 50, x r = 50, μ = γ = κ = ν = 1 . The initial condition is

ϕ (x) = \frac{\sqrt{6}}{3} {normalsech}^{2} (\frac{x}{3})

.…”

Section: Numerical Experimentssupporting

confidence: 91%

“…For the sake of numerical comparison, recently, Berikelashvili and Mirianashvili [42] proposed a linearized difference scheme for the generalized BBMB equation by taking the average of two layers of (1.1) at k − 1 and k + 1 as follows

δ_{t} u_{i}^{\frac{1}{2}} - {μ δ}_{t} δ_{x}^{2} u_{i}^{\frac{1}{2}} + γ ψ {((),, u^{0}, u^{\frac{1}{2}})}_{i} + κ Δ_{x} u_{i}^{\frac{1}{2}} - {ν δ}_{x}^{2} u_{i}^{\frac{1}{2}} = 0, 1 ⩽ i ⩽ m - 1,

\begin{array}{l} Δ_{t} u_{i}^{k} - μ Δ_{t} δ_{x}^{2} u_{i}^{k} + γ ψ {(u^{k}, u^{\overset{true‾}{k}})}_{i} + κ Δ_{x} u_{i}^{\overset{true‾}{k}} - {ν δ}_{x}^{2} u_{i}^{\overset{true‾}{k}} = 0, \\ 1 ⩽ i ⩽ m - 1, 1 ⩽ k ⩽ n - 1, \end{array}

u_{i}^{0} = φ (x_{i}), 1 ⩽ i ⩽ m - 1,

u_{0}^{k} = 0, u_{m}^{k} = 0, 0 ⩽ k ⩽ n .

They also delicately proved the convergence of (6.1)–(6.4) with the convergence order …”

Section: Numerical Experimentsmentioning

confidence: 99%

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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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ϕ (x) = \frac{\sqrt{6}}{3} {normalsech}^{2} (\frac{x}{3})

.…”

Section: Numerical Experimentssupporting

confidence: 91%

δ_{t} u_{i}^{\frac{1}{2}} - {μ δ}_{t} δ_{x}^{2} u_{i}^{\frac{1}{2}} + γ ψ {((),, u^{0}, u^{\frac{1}{2}})}_{i} + κ Δ_{x} u_{i}^{\frac{1}{2}} - {ν δ}_{x}^{2} u_{i}^{\frac{1}{2}} = 0, 1 ⩽ i ⩽ m - 1,

\begin{array}{l} Δ_{t} u_{i}^{k} - μ Δ_{t} δ_{x}^{2} u_{i}^{k} + γ ψ {(u^{k}, u^{\overset{true‾}{k}})}_{i} + κ Δ_{x} u_{i}^{\overset{true‾}{k}} - {ν δ}_{x}^{2} u_{i}^{\overset{true‾}{k}} = 0, \\ 1 ⩽ i ⩽ m - 1, 1 ⩽ k ⩽ n - 1, \end{array}

u_{i}^{0} = φ (x_{i}), 1 ⩽ i ⩽ m - 1,

u_{0}^{k} = 0, u_{m}^{k} = 0, 0 ⩽ k ⩽ n .

They also delicately proved the convergence of (6.1)–(6.4) with the convergence order …”

Section: Numerical Experimentsmentioning

confidence: 99%

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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“…Although the Burgers equation is unphysical [12], it is nevertheless relevant to various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow [5,10]. An important role in mathematical physics is played by various modifications of the Burgers equation, such as the generalized Burgers equation [6,14,21], Burgers-Fisher equation [34], Korteweg-de Vries-Burgers equation [32,39], Rosenau-Burgers [22,36] and others. The Korteweg-de Vries-Burgers equation is obtained when in the models describing propagation of undular bores in shallow water and in fluids containing gas bubbles a smoothing effect is added and produces a third phenomenon, dissipation (second-order term) [3,17].…”

Section: Introductionmentioning

confidence: 99%

On an Inverse Problem With an Integral Overdetermination Condition for the Burgers Equation

Yergaliyev

Jenaliyev²,

Romankyzy³

et al. 2023

JMMCS

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In this paper we consider one inverse problem for the Burgers equation with an integral overdetermination and periodic boundary conditions in a domain that is trapezoid. Using an integral overdetermination, boundary and initial conditions, we reduce the inverse problem to the study of an already direct initial boundary value problem for the loaded Burgers equation. Next, we use a one-to-one transformation of independent variables to move from a trapezoid to a rectangular domain, where we study an auxiliary problem, for which the methods of Faedo-Galerkin, a priori estimates and functional analysis have been proved a theorem on its unique solvability in Sobolev classes. Note that the obtained a priori estimates are uniform with respect to the summation index of the approximate solution and do not depend on time. Further, on the basis of this theorem, due to the correspondence of spaces, theorems on the unique solvability of the original inverse problem are proved. Also, for the selected initial data, the paper presents graphs of the initial-boundary problem for the loaded Burgers equation and the desired function of the inverse problem, which together constitute the solution of the original inverse problem.

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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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On the convergence of difference schemes for the generalized BBM–Burgers equation

Cited by 3 publications

References 10 publications

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

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

On an Inverse Problem With an Integral Overdetermination Condition for the Burgers Equation

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

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