Some new semi-implicit finite difference schemes for numerical solution of Burgers equation

Rahman, Kaysar; Helil, Nurmamat; Yimin, Rahmatjan

doi:10.1109/iccasm.2010.5622119

Cited by 19 publications

(15 citation statements)

References 19 publications

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“…Problem 5.10 : We consider the initial and boundary conditions for this example as given in (Asaithambi, 2010; Rahman et al , 2010; Mittal and Jain, 2012): …”

Section: Numerical Experiments and Discussionmentioning

confidence: 99%

Meshfree algorithms based on radial basis functions for numerical simulation and to capture shocks behavior of Burgers’ type problems

Jiwari

Kumar

Mittal

2019

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Purpose The purpose of this paper is to develop two meshfree algorithms based on multiquadric radial basis functions (RBFs) and differential quadrature (DQ) technique for numerical simulation and to capture the shocks behavior of Burgers’ type problems. Design/methodology/approach The algorithms convert the problems into a system of ordinary differential equations which are solved by the Runge–Kutta method. Findings Two meshfree algorithms are developed and their stability is discussed. Numerical experiment is done to check the efficiency of the algorithms, and some shock behaviors of the problems are presented. The proposed algorithms are found to be accurate, simple and fast. Originality/value The present algorithms LRBF-DQM and GRBF-DQM are based on radial basis functions, which are new for Burgers’ type problems. It is concluded from the numerical experiments that LRBF-DQM is better than GRBF-DQM. The algorithms give better results than available literature.

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“…Problem 5.10 : We consider the initial and boundary conditions for this example as given in (Asaithambi, 2010; Rahman et al , 2010; Mittal and Jain, 2012): …”

Section: Numerical Experiments and Discussionmentioning

confidence: 99%

Meshfree algorithms based on radial basis functions for numerical simulation and to capture shocks behavior of Burgers’ type problems

Jiwari

Kumar

Mittal

2019

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“…So far various numerical algorithms such as finite difference and cubic spline finite element methods [6], the group-explicit method [7], the generalized boundary element approach [8], quartic B-splines collocation method [9], quadratic B-splines finite element method [10], finite element method [11], spectral method [32], fourthorder finite difference method [12], a novel numerical scheme [13], explicit and exactexplicit finite difference methods [14], automatic differentiation method [15], Galerkin finite element method [16], cubic B-splines collocation method [17], spectral collocation method [18], Polynomial based differential quadrature method [19], quartic B-splines differential quadrature method [20], least-squares quadratic B-splines finite element method [21], implicit fourth-order compact finite difference scheme [22], some implicit methods [23], variational iteration method [24], homotopy analysis method [25], differential transform method and the homotopy analysis method [26], a numerical method based on Crank-Nicolson [27], modified cubic B-splines collocation method [28] ,differential quadrature method [29][30][31]46,47], some new semi-implicit finite difference schemes [33], Haar wavelet quasilinearization approach [34] etc. have been developed for the numerical solutions of Burgers' equation.…”

Section: Introductionmentioning

confidence: 99%

A hybrid numerical scheme for the numerical solution of the Burgers’ equation

Jiwari

2015

Computer Physics Communications

140

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“…Figure 21 exhibit that the shock wave exist in the range x ∈ (0.9, 1) at time t = 0.33 for planar geometry ( n = 0). The results are compared with Öziş et al (2003), Kutulay et al (1999), Kutluay et al (2004), Kadalbajoo and Awasthi (2006), Rahman et al (2010) and Jiwari (2012) for time t ≤ 15. It is observed that the numerical solutions by using the proposed method are in good agreement with exact solution for modest values of v and exhibit the better solutions for refined time step Δt.…”

Section: Numerical Experiments and Discussionmentioning

confidence: 99%

Shock waves analysis of planar and non planar nonlinear Burgers’ equation using Scale-2 Haar wavelets

Pandit

Kumar

Mohapatra

2017

HFF

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Purpose This paper aims to find the numerical solution of planar and non-planar Burgers’ equation and analysis of the shock behave. Design/methodology/approach First, the authors discritize the time-dependent term using Crank–Nicholson finite difference approximation and use quasilinearization to linearize the nonlinear term then apply Scale-2 Haar wavelets for space integration. After applying this scheme on partial differential, the equation transforms into a system of algebraic equation. Then, the system of equation is solved using Gauss elimination method. Findings Present method is the extension of the method (Jiwari, 2012). The numerical solutions using Scale-2 Haar wavelets prove that the proposed method is reliable for planar and non-planar nonlinear Burgers’ equation and yields results better than other methods and compatible with the exact solutions. Originality/value The numerical results for non-planar Burgers’ equation are very sparse. In the present paper, the authors identify where the shock wave and discontinuity occur in planar and non-planar Burgers’' equation.

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Some new semi-implicit finite difference schemes for numerical solution of Burgers equation

Cited by 19 publications

References 19 publications

Meshfree algorithms based on radial basis functions for numerical simulation and to capture shocks behavior of Burgers’ type problems

Meshfree algorithms based on radial basis functions for numerical simulation and to capture shocks behavior of Burgers’ type problems

A hybrid numerical scheme for the numerical solution of the Burgers’ equation

Shock waves analysis of planar and non planar nonlinear Burgers’ equation using Scale-2 Haar wavelets

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