Browse > Article
http://dx.doi.org/10.5666/KMJ.2012.52.3.327

A New Time Stepping Method for Solving One Dimensional Burgers' Equations  

Piao, Xiang Fan (Department of Mathematics, Kyungpook National University)
Kim, Sang-Dong (Department of Mathematics, Kyungpook National University)
Kim, Phil-Su (Department of Mathematics, Kyungpook National University)
Kim, Do-Hyung (Department of Physics, Kyungpook National University)
Publication Information
Kyungpook Mathematical Journal / v.52, no.3, 2012 , pp. 327-346 More about this Journal
Abstract
In this paper, we present a simple explicit type numerical method for discretizations in time for solving one dimensional Burgers' equations. The proposed method does not need an iteration process that may be required in most implicit methods and have good convergence and efficiency in computational sense compared to other known numerical methods. For evidences, several numerical demonstrations are also provided.
Keywords
Error corrected Euler method; Pseudo-Spectral method;
Citations & Related Records
연도 인용수 순위
  • Reference
1 R. C. Mittal, R. Jiwari, Differential Quadrature Method for Two-Dimensional Burgers' equations, Int. J. Comput. Methods Eng. Sci. Mech., 10(2009), 450-459.   DOI   ScienceOn
2 A. R. Mitchell, D. F. Griffiths, The finite difference method in partial differential equations, John Wiley & Sons, New York, 1980.
3 T. Ozis, E. N. Aksan and A. Ozdes, A finite element approach for solution of Burgers' equation, Applied Math. and Comput., 139(2003), 417-428.   DOI   ScienceOn
4 H. Ramos, J. Vigo-Aguiar, A fourth-order Runge-Kutta method based on BDF-type Chebyshev approximations, J. Comput. Appl. Math., 204(2007), 124-136.   DOI   ScienceOn
5 L. N. Trefethen, Spectral Methods in MATLAB, Software Environ. Tools 10, SIAM, Philadelphia, 2000.
6 Y. Wu, X. H. Wu, Linearized and rational approximation method for solving nonlinear Burgers' equation, Int. J. Numer. Methods Fluids, 45(2004), 509-525.   DOI   ScienceOn
7 M. Xu, R. H. Wang, J. H. Zhang and Q. Fang, A novel numerical scheme for solving Burgers' equation, Applied Math. and Comput., 217(2011), 4473-4482.   DOI   ScienceOn
8 L. Zhang, J. Ouyang, X. Wang and X. Zhang, Varational multiscale element-free Galerkin method for 2D Burgers' equation, J. Comput. Phys., 229(2010), 7147-7161.   DOI   ScienceOn
9 I. A. Hassanien, A. A. Salama and H. A. Hosham Fourth-order finite difference method for solving Burgers' equation, Applied Math. and Comput., 170(2005), 781-800.   DOI   ScienceOn
10 B. M. Herbst, S. W. Schoombie, D. F. Griffiths and A. R. Mitchell, Generalized Petrov-Galerkin methods for the numerical solution of Burgers' equation, Int. J. Numer. Methods Eng., 20(1984), 1273-1289.   DOI   ScienceOn
11 R. S. Hirsh, Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique, J. Comput. Phys., 19(1975), 90-105.   DOI   ScienceOn
12 A. N. Hrymak, G. J. Mcrae and A. W. Westerberg, An implementation of a moving finite element method, J. Comput. Phys., 63(1986), 168-190.   DOI   ScienceOn
13 P. Z. Hunag, A. Abduwali, The modified local Crank-Nicolson method for one- and two-dimensional Burgers' equations, Compu. Math. Appl., 59(2010), 2452-2463.   DOI   ScienceOn
14 A. H. Khater, R. S. Temsah and M. M. Hassan, A Chebyshev spectral collocation method for solving Burgers'-type equations, J. Comput. Appl. Math., 222(2008), 333- 350.   DOI   ScienceOn
15 P. Kim, X. Piao and S. Kim, An error corrected Euler method for solving stiff problems based on Chebyshev collocation, SIAM J. Numer. Anal., 49(2011), 2211-2230.   DOI   ScienceOn
16 K. Black, A spectral element technique with a local spectral basis, SIAM J. Sci. Comput., 18(1997), 355-370.   DOI   ScienceOn
17 S. Kutluay, A. R. Bahadir and A. Ozdes, Numerical solution of one-dimensional Burgers equation: explicit and exact-explicit finite difference methods, J. Comput. and Applied Math., 103(1999), 251-261.   DOI   ScienceOn
18 S. Kutluay, A. Esen and I. Dag, Numerical solutions of the Burgers' equation by the least-squares quadratic B-spline finite element method, J. Comput. Appl. Math., 167(2004), 21-33.   DOI   ScienceOn
19 M. J. Lighthill, 'Viscosity Effects in Sound Waves of Finite Amplitude', In: Surveys in Mechanics, ed. by G.K. Bauchelor, R. Davies, Combridge Univ. Press, 1956.
20 D. T. Blackstock, Convergence of the Keck-Boyer perturbation solution for plane waves of finite amplitude in vicous fluid, J. Acoust. Soc. Am., 39(1966), 411-413.   DOI
21 N. Bressan, A. Quarteroni, An implicit/explicit spectral method for Burgers' equation, Calcolo, 23(1987), 265-284.
22 J. M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 1(1948), 171-199.   DOI
23 J. Caldwell, P. Wanless and A. E. Cook, A finite element approach to Burgers' equation, Appl. Math. Modelling, 5(1981), 189-193.   DOI   ScienceOn
24 C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin Heidelberg, 2006.
25 H. Chen, Z. Jiang, A characteristics mixed finite element method for Burgers' equation, J. Appl. Math. Comput., 15(2004), 29-51.   DOI
26 G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3(1963), 27-43.   DOI
27 I. Christie, A. R. Mitchell, Upwinding of high order Galerkin methods in conductionconvection problems, Int. J. Numer. Methods Eng., 12(1978), 1764-1771.   DOI   ScienceOn
28 M. Ciment, S. H. Leventhal and B. C. Weinberg, The operator compact implicit method for parabolic equations, J. Comput. Phys., 28(1978), 135-166.   DOI   ScienceOn
29 J. D. Cole, On a quasi-linear parabolic equation occuring in aerodynamics, Quart. Appl. Math., IX, (1951), 225-236.
30 M. O. Deville, P. F. Fischer and E. H. Mund, High-order methods for incompressible fluid flow, Cambridge University Press, New York, 2002.
31 M. Berzins, Global error estimation in the methods of lines for parabolic equations, SIAM J. Sci. Statist. Comput., 19(4)(1988), 687-701.
32 A. H. A. Ali, G. A. Gardner, L. R. T. Gardner, A collocation solution for Burgers' equation using cubic B-spline finite elements, Comput. Methods Appl. Mech. Engrg., 100(1992), 325-337.   DOI   ScienceOn
33 A. R. Bahadir, A fully implicit finite-difference scheme for two-dimensional Burgers' equations, Appl. Math. Comput., 137(2003), 131-137.   DOI   ScienceOn