DOI QR코드

DOI QR Code

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)
  • 투고 : 2011.04.06
  • 심사 : 2011.09.23
  • 발행 : 2012.09.23

초록

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.

키워드

참고문헌

  1. 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. https://doi.org/10.1016/0045-7825(92)90088-2
  2. A. R. Bahadir, A fully implicit finite-difference scheme for two-dimensional Burgers' equations, Appl. Math. Comput., 137(2003), 131-137. https://doi.org/10.1016/S0096-3003(02)00091-7
  3. M. Berzins, Global error estimation in the methods of lines for parabolic equations, SIAM J. Sci. Statist. Comput., 19(4)(1988), 687-701.
  4. K. Black, A spectral element technique with a local spectral basis, SIAM J. Sci. Comput., 18(1997), 355-370. https://doi.org/10.1137/S1064827594268713
  5. 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. https://doi.org/10.1121/1.1909911
  6. N. Bressan, A. Quarteroni, An implicit/explicit spectral method for Burgers' equation, Calcolo, 23(1987), 265-284.
  7. J. M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech., 1(1948), 171-199. https://doi.org/10.1016/S0065-2156(08)70100-5
  8. J. Caldwell, P. Wanless and A. E. Cook, A finite element approach to Burgers' equation, Appl. Math. Modelling, 5(1981), 189-193. https://doi.org/10.1016/0307-904X(81)90043-3
  9. C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin Heidelberg, 2006.
  10. H. Chen, Z. Jiang, A characteristics mixed finite element method for Burgers' equation, J. Appl. Math. Comput., 15(2004), 29-51. https://doi.org/10.1007/BF02935745
  11. I. Christie, A. R. Mitchell, Upwinding of high order Galerkin methods in conductionconvection problems, Int. J. Numer. Methods Eng., 12(1978), 1764-1771. https://doi.org/10.1002/nme.1620121113
  12. M. Ciment, S. H. Leventhal and B. C. Weinberg, The operator compact implicit method for parabolic equations, J. Comput. Phys., 28(1978), 135-166. https://doi.org/10.1016/0021-9991(78)90031-1
  13. J. D. Cole, On a quasi-linear parabolic equation occuring in aerodynamics, Quart. Appl. Math., IX, (1951), 225-236.
  14. G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3(1963), 27-43. https://doi.org/10.1007/BF01963532
  15. M. O. Deville, P. F. Fischer and E. H. Mund, High-order methods for incompressible fluid flow, Cambridge University Press, New York, 2002.
  16. 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. https://doi.org/10.1016/j.amc.2004.12.052
  17. 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. https://doi.org/10.1002/nme.1620200708
  18. R. S. Hirsh, Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique, J. Comput. Phys., 19(1975), 90-105. https://doi.org/10.1016/0021-9991(75)90118-7
  19. 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. https://doi.org/10.1016/0021-9991(86)90090-2
  20. 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. https://doi.org/10.1016/j.camwa.2009.08.069
  21. 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. https://doi.org/10.1016/j.cam.2007.11.007
  22. 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. https://doi.org/10.1137/100808691
  23. 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. https://doi.org/10.1016/S0377-0427(98)00261-1
  24. 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. https://doi.org/10.1016/j.cam.2003.09.043
  25. 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.
  26. R. C. Mittal, R. Jiwari, Differential Quadrature Method for Two-Dimensional Burgers' equations, Int. J. Comput. Methods Eng. Sci. Mech., 10(2009), 450-459. https://doi.org/10.1080/15502280903111424
  27. A. R. Mitchell, D. F. Griffiths, The finite difference method in partial differential equations, John Wiley & Sons, New York, 1980.
  28. 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. https://doi.org/10.1016/S0096-3003(02)00204-7
  29. 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. https://doi.org/10.1016/j.cam.2006.04.033
  30. L. N. Trefethen, Spectral Methods in MATLAB, Software Environ. Tools 10, SIAM, Philadelphia, 2000.
  31. Y. Wu, X. H. Wu, Linearized and rational approximation method for solving nonlinear Burgers' equation, Int. J. Numer. Methods Fluids, 45(2004), 509-525. https://doi.org/10.1002/fld.714
  32. 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. https://doi.org/10.1016/j.amc.2010.10.050
  33. 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. https://doi.org/10.1016/j.jcp.2010.06.004