ZHAN Jie-min, LI Yok-sheng. Generalized Finite Spectral Method for 1D Burgers and KdV Equations[J]. Applied Mathematics and Mechanics, 2006, 27(12): 1431-1438.
Citation: ZHAN Jie-min, LI Yok-sheng. Generalized Finite Spectral Method for 1D Burgers and KdV Equations[J]. Applied Mathematics and Mechanics, 2006, 27(12): 1431-1438.

Generalized Finite Spectral Method for 1D Burgers and KdV Equations

  • Received Date: 2005-05-15
  • Rev Recd Date: 2006-06-30
  • Publish Date: 2006-12-15
  • A generalized finite spectral method is proposed. The method is of high-order accuracy. To attain high accuracy in time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme was used. To avoid numerical oscillations caused by the dispersion term in the KdV equation, two numerical techniques were introduced to improve the numerical stability. The Legendre, Chebyshev and Hermite polynomials were used as the basis functions. The proposed numerical scheme is validated by applications to the Burgers equation (nonlinear convection-diffusion problem) and KdV equation(single solitary and 2-solitary wave problems), where analytical solutions are available for comparison. Numerical results agree very well with the corresponding analytical solutions in all cases.
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  • [1]
    de Boussinesq J. Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal en communiquant au liquide continu dans ce canal des vitesses sensiblement pareilles de la surface au fond[J].J Math Pures et Appliquées,1872,17(2):55—108.
    Korteweg D J, de Vries G. On the change of form of long waves advancing in a rectangular canal and on a new type of long sationary waves[J].Philosophical Magazine, 5th Series,1895,36:422—443.
    Pego R L,Smereka P, Weinstein M I. Oscillatory instability of traveling waves for a KdV-Burgers equation[J].Physica D,1993,67(1/3):45—65. doi: 10.1016/0167-2789(93)90197-9
    Ge H X, Dai S Q,Dong L Y,et al.Stabilization effect of traffic flow in an extended car-following model based on an intelligent transportation system application[J].Physical Review E,2004,70(6):Art.No.066134 Part 2.
    Zhang C Y, Tan H L, Liu M R,et al.A lattice Boltzmann model and simulation of KdV-Burgers equation[J].Communications in Theoretical Physics,2004,42(2):281—284.
    Kaya D.On the solution of a Korteweg-de Vries like equation by the decomposition method[J].International Journal of Computer Mathematics,1999,72(4):531—539. doi: 10.1080/00207169908804874
    Kaya D. Solitary-wave solutions for compound KdV-type and compound KdV-Burgers-type equations with nonlinear terms of any order[J].Applied Mathematics and Computation,2004,152(3):709—720. doi: 10.1016/S0096-3003(03)00589-7
    Patera A T. A spectral element method for fluid-dynamics -laminar-flow in a channel expansion[J].Journal of Computational Physics,1984,54(3):468—488. doi: 10.1016/0021-9991(84)90128-1
    Ghaddar N K, Karniadakis G E, Patera A T. A conservative isoparametric spectral element method for forced convection: Application to fully developed flow in periodic geometries[J].Numer Heat Transfer,1986,9(3):277—300.
    Giraldo F X. Strong and weak Lagrange-Galerkin spectral element methods for the shallow water equations[J].Computers and Mathematics With Applications,2003,45(1/3):97—121. doi: 10.1016/S0898-1221(03)80010-X
    WANG Jian-ping.Non-periodic Fourier tansform and finite spectral method[A].In:Sixth Intere Symposium in CFD[C].Nevada:Lake Tahoe,1995,1339—1344.
    WANG Jian-ping.Finite spectral method based on non-periodic Fourier transform[J].Computers and Fluids,1998,27(5/6):639—644. doi: 10.1016/S0045-7930(97)00056-X
    沈盂育,张增产,李海东. 高精度三点有限谱方法[J].清华大学学报(自然科学版),1997,37(8):52—54.
    Su C H, Gardner C S.Derivation of the Korteweg-de Vries and Burgers equation[J].J Math Phys,1969,10(3):536—539. doi: 10.1063/1.1664873
    Li Y S, Zhan J M.Boussinesq-type model with boundary-fitted coordinate system[J].Journal of Waterway Port Coastal and Ocean Engineering,ASCE,2001,127(3):152—160. doi: 10.1061/(ASCE)0733-950X(2001)127:3(152)
    Beji S, Nadaoka K.A formal derivation and numerical modelling of the improved Boussinesq equations for varying depth[J].Ocean Engineering,1996,23(8):691—704. doi: 10.1016/0029-8018(96)84408-8
    Press W H, Flannery B P, Teukolsky S A,et al.Numerical Recipes[M].New York:Cambridge University Press, 1989,569—572.
    Wei G, Kirby J T.Time-dependent numerical code for extended Boussinesq equations[J].Journal of Waterway, Port, Coastal, and Ocean Engineering, ASCE,1995,121(5):251—260. doi: 10.1061/(ASCE)0733-950X(1995)121:5(251)
    Dodd R K,Eilbeck J C,Gibbon J D,et al.Solitons and Nonlinear Wave Equations[M].New York :Academic Press, 1984.
    Li P W. On the numerical study of the KdV equation by the semi-implicit and Leap-frog method[J].Computer Physics Communications,1995,88(2/3):121—127. doi: 10.1016/0010-4655(95)00060-S
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