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Hermite spectral method for Long-Short wave equations

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  • We are concerned with the initial boundary value problem of the Long-Short wave equations on the whole line. A fully discrete spectral approximation scheme is structured by means of Hermite functions in space and central difference in time. A priori estimates are established which are crucial to study the numerical stability and convergence of the fully discrete scheme. Then, unconditionally numerical stability is proved in a space of $H^1({\Bbb R})$ for the envelope of the short wave and in a space of $L^2({\Bbb R})$ for the amplitude of the long wave. Convergence of the fully discrete scheme is shown by the method of error estimates. Finally, numerical experiments are presented and numerical results are illustrated to agree well with the convergence order of the discrete scheme.

    Mathematics Subject Classification: 65M12, 35B45, 76M22.


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  • Figure 1.  Errors for the Short wave

    Figure 2.  Errors for the Long wave

    Table 1.  Errors and convergence rates for the Short wave

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    Table 2.  Errors and convergence rates for the Long wave

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    Table 3.  Errors for both the Short wave and Long wave

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  •   Q. S. Chang , Y. S. Wong  and  C. K. Lin , Numerical computations for long-wave short-wave interaction equations in semi-classical limit, J. Comput. Phys., 227 (2008) , 8489-8507.  doi: 10.1016/j.jcp.2008.05.015.
      V. D. Djordjevic  and  L. G. Redekop , On two-dimensional packets of capillary-gravity waves, J. Fluid. Mech., 79 (1977) , 703-714.  doi: 10.1017/S0022112077000408.
      B. L. Guo , The global solution for one class of the system of LS nonlinear wave interaction, J. Math. Res., & Exposition, 7 (1987) , 69-76. 
      B. Y. Guo , Error estimation for Hermite spectral method for nonlinear partial differential equations, Math. Comput., 68 (1999) , 1067-1078.  doi: 10.1090/S0025-5718-99-01059-5.
      B. Y. Guo , J. Shen  and  C. L. Xu , Spectral and pseudospectral approximations using Hermite functions: application to the Dirac equation, Adv. Comput. Math., 19 (2003) , 35-55.  doi: 10.1023/A:1022892132249.
      X. Luo  and  S. S. Yau , Hermite spectral method to 1-D Forward Kolmogorov Equation and its application to nonlinear filtering problems, IEEE T. Automat Contr., 58 (2013) , 2495-2507.  doi: 10.1109/TAC.2013.2259975.
      H. P. Ma  and  W. W. Sun , Optimal error estimates of the Legendre-Petrov-Galerkin method for the Korteweg-de Vries equaiton, SIAM J. Numer. Anal., 39 (2001) , 1380-1394.  doi: 10.1137/S0036142900378327.
      H. P. Ma , W. W. Sun  and  T. Tang , Hermite spectral methods with a time-dependent scaling for parabolic equations in unbounded domains, SIAM J. Numer. Anal., 43 (2005) , 58-75.  doi: 10.1137/S0036142903421278.
      D. R. Nicholson  and  M. V. Goldman , Damped nonlinear Schröinger equation, Phys. Fluid., 19 (1976) , 1621-1625.  doi: 10.1063/1.861368.
      K. Nishikawa , H. Hojo , K. Mima  and  H. Ikezi , Coupled nonlinear electron-plasma and ion acoustic waves, Phys. Rev. Lett., 33 (1974) , 148-151. 
      A. Quarteroni and A. Vall, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin, 1994.
      A. Rashid , The pseudo-spectral collocation method for resonant long-short nonlinear wave interaction, Georgian. Math. J., 13 (2006) , 143-152. 
      A. Rashid  and  S. Akram , Convergence of Fourier spectral method for resonant long-short nonlinear wave interaction, Appl. Math-Czech., 55 (2010) , 337-350.  doi: 10.1007/s10492-010-0025-5.
      M. Tsutsumi  and  S. Hatano , Well-posedness of the Cauchy problem for Benney's first equations of long-wave-short-wave interactions, Funkcial. Ekvac., 37 (1994) , 289-316. 
      M. Tsutsumi  and  S. Hatano , Well posedness of the Cauchy problem for the long wave-short wave resonance equations, Nonlinear. Anal., 22 (1994) , 155-171.  doi: 10.1016/0362-546X(94)90032-9.
      X. M. Xiang  and  Z. Q. Wang , Generalized Hermite spectral method and its applications to problems in unbounded domains, SIAM J. Numer. Anal., 48 (2010) , 1231-1253.  doi: 10.1137/090773581.
      C. L. Xu  and  B. Y. Guo , Hermite spectral and pseudospectral methods for nonlinear partial differential equations in multiple dimensions, Comput. App. Math., 22 (2003) , 167-193.  doi: 10.1590/S0101-82052003000200002.
      C. Y. Zhang  and  B. Y. Guo , Generalized Hermite spectral method matching asymptotic behaviors, J. Comput. Appl. Math., 255 (2014) , 616-634.  doi: 10.1016/j.cam.2013.06.018.
      F. Y. Zhang  and  X. M. Xiang , Pseudospectral method for a class of system of LS wave interaction, Numer. Math. Nanjing., 12 (1990) , 199-214. 
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