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) |
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 |