Dynamics and synchronization of numerical solutions of the Burgers equation

Basto, Mário; Semião, Viriato; Calheiros, Francisco

doi:10.1016/j.cam.2009.05.003

Cited by 23 publications

(13 citation statements)

References 33 publications

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“…For instance, the two‐dimensional unsteady nonlinear coupled Burgers' equations are such type of PDEs. Burgers' equations occur in a wide area of applied mathematics such as heat conduction, modeling of dynamics, acoustic wave, turbulent fluids, and continuous stochastic processes . Numerical analysis of Burgers' equations has attracted attention during the last decades, and it represents an active filed of research to develop fast and efficient numerical schemes in the approximate solutions of such equations.…”

Section: Introductionmentioning

confidence: 99%

“…with initial condition u(x, , 0) = u 0 (x, ), v(x, , 0) = v 0 (x, ), (x, ) ∈ Ω (2) and boundary condition…”

mentioning

confidence: 99%

“…In this paper, we analyze the three-level time-split MacCormack procedure for the two-dimensional evolutionary nonlinear coupled Burgers' equations (1) subject to the initial and boundary conditions (2) and (3), respectively. Specifically, this work is motivated by: (a) for low Reynolds numbers, the method is fast, efficient (explicit, stable, second-order accurate in time, and fourth-order convergent in space), and easy to implement under the time step requirement 2k Rh 2 ≤ 1; meanwhile, for high Reynolds numbers and under the time step constraint k 3 4 h ≤ 1, the algorithm is efficient (explicit, stable, and second-order accurate in both time and space).…”

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confidence: 99%

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An efficient three‐level explicit time‐split scheme for solving two‐dimensional unsteady nonlinear coupled Burgers' equations

Ngondiep

2019

Numerical Methods in Fluids

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Summary This work deals with the numerical solutions of two‐dimensional viscous coupled Burgers' equations with appropriate initial and boundary conditions using a three‐level explicit time‐split MacCormack approach. In this technique, the differential operators split the two‐dimensional problem into two pieces so that the two‐step explicit MacCormack scheme can be easily applied to each subproblem. This reduces the computational cost of the algorithm. For low Reynolds numbers, the proposed method is second‐order accurate in time and fourth‐order convergent in space, whereas it is second‐order convergent in both time and space for high Reynolds numbers problems. This observation shows the utility and efficiency of the considered method compared with a broad range of numerical schemes widely studied in the literature for solving the two‐dimensional time‐dependent nonlinear coupled Burgers' equations. A large set of numerical examples that confirm the theoretical results are presented and critically discussed.

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Section: Introductionmentioning

confidence: 99%

“…with initial condition u(x, , 0) = u 0 (x, ), v(x, , 0) = v 0 (x, ), (x, ) ∈ Ω (2) and boundary condition…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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An efficient three‐level explicit time‐split scheme for solving two‐dimensional unsteady nonlinear coupled Burgers' equations

Ngondiep

2019

Numerical Methods in Fluids

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“…In fluid dynamics, Burgers equation can be considered as a model equation or approach to the NavierStokes equation, since the main terms are remained, and it can be used to study the turbulence, shock wave, soliton etc. There are a very rich variety of nonlinear phenomena in it [2][3][4][5]. In particular, one of the phenomena is the sharp jumping or discontinuities as the Reynolds becomes higher or the viscosity coefficient is lower, and such phenomenon is relevant to the shock wave which is encountered frequently in the aircraft with supersonic speed.…”

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confidence: 99%

Saddle-node bifurcations in burgers equation as shock wave occurrence

Zhang

Liu

Ren

et al. 2012

2012 IEEE 4th International Conference on Nonlinear Science and Complexity (NSC)

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The nature of the shock wave, governed by Burgers equation, is presented from viewpoint of saddlenode bifurcations. First, the inviscid Burgers equation is studied in detail, and the solution of the system with a certain initial condition is obtained. The solution in vector form is reduced into a Map in order to investigate the stability and bifurcation in the system. It has been proved that there exists a thin spatial zone where a saddle-node bifurcation occurs as time goes on, and the velocity of the fluid behaves as jumping, namely, the characteristic of shock wave. Further, the period-doubling bifurcation is captured, that means there exist multiple states as time increases, and the complicated spatio-temporal pattern is formatted. In addition to above, the viscous Burgers equation is further studied to extend to dissipative systems. By traveling wave transformation, the governing equation is reduced into an ordinary differential equation. Then, the instability or bifurcation condition is obtained, and it is proved that there are three singular points in the system as the bifurcation condition is satisfied. More, the velocity distribution is obtained with a certain initial condition. The results show that the discontinuity resulting from saddlenode bifurcation is removed with the introduction of viscousity, and another kind of velocity change with strong gradient is obtained. However, the change of velocity is continuous with sharp slopes. As a conclusion, all results can provide a fundamental understanding of the nonlinear phenomena relevant to shock wave and other complicated nonlinear phenomena, from viewpoint of nonlinear dynamics.

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“…It occurs in various areas of applied mathematics, such as modeling of dynamics, heat conduction, and acoustic waves [13][14][15]. Numerical techniques for the Burgers' equation usually fall into the following classes: finite differences, finite elements, and spectral methods [16].…”

Section: Introductionmentioning

confidence: 99%

Quasi-Interpolation Operators for Bivariate Quintic Spline Spaces and Their Applications

Zhu

Hou

et al. 2017

MCA

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Splines and quasi-interpolation operators are important both in approximation theory and applications. In this paper, we construct a family of quasi-interpolation operators for the bivariate quintic spline spaces S 3 5 p∆ p2q mn q. Moreover, the properties of the proposed quasi-interpolation operators are studied, as well as its applications for solving the two-dimensional Burgers' equation and image reconstruction. Some numerical examples show that these methods, which are easy to implement, provide accurate results.

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Dynamics and synchronization of numerical solutions of the Burgers equation

Cited by 23 publications

References 33 publications

An efficient three‐level explicit time‐split scheme for solving two‐dimensional unsteady nonlinear coupled Burgers' equations

An efficient three‐level explicit time‐split scheme for solving two‐dimensional unsteady nonlinear coupled Burgers' equations

Saddle-node bifurcations in burgers equation as shock wave occurrence

Quasi-Interpolation Operators for Bivariate Quintic Spline Spaces and Their Applications

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