February  2019, 24(2): 941-964. doi: 10.3934/dcdsb.2018255

Hermite spectral method for Long-Short wave equations

1. 

School of Mathematics and Systems Science & LMIB, Beihang University, Beijing 100191, China

2. 

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

3. 

Department of Mathematics, University of Texas-Rio Grande Valley, Edinburg, Texas 78539, USA

* Corresponding author: zhaosheng.feng@utrgv.edu; fax: (956) 665-5091

Received  January 2016 Revised  July 2018 Published  October 2018

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.

Citation: Shujuan Lü, Zeting Liu, Zhaosheng Feng. Hermite spectral method for Long-Short wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 941-964. doi: 10.3934/dcdsb.2018255
References:
[1]

Q. S. ChangY. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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B. Y. GuoJ. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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H. P. MaW. 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.  Google Scholar

[9]

D. R. Nicholson and M. V. Goldman, Damped nonlinear Schröinger equation, Phys. Fluid., 19 (1976), 1621-1625.  doi: 10.1063/1.861368.  Google Scholar

[10]

K. NishikawaH. HojoK. Mima and H. Ikezi, Coupled nonlinear electron-plasma and ion acoustic waves, Phys. Rev. Lett., 33 (1974), 148-151.   Google Scholar

[11]

A. Quarteroni and A. Vall, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin, 1994.  Google Scholar

[12]

A. Rashid, The pseudo-spectral collocation method for resonant long-short nonlinear wave interaction, Georgian. Math. J., 13 (2006), 143-152.   Google Scholar

[13]

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.  Google Scholar

[14]

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.   Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

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.   Google Scholar

show all references

References:
[1]

Q. S. ChangY. 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.  Google Scholar

[2]

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.  Google Scholar

[3]

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.   Google Scholar

[4]

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.  Google Scholar

[5]

B. Y. GuoJ. 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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

H. P. MaW. 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.  Google Scholar

[9]

D. R. Nicholson and M. V. Goldman, Damped nonlinear Schröinger equation, Phys. Fluid., 19 (1976), 1621-1625.  doi: 10.1063/1.861368.  Google Scholar

[10]

K. NishikawaH. HojoK. Mima and H. Ikezi, Coupled nonlinear electron-plasma and ion acoustic waves, Phys. Rev. Lett., 33 (1974), 148-151.   Google Scholar

[11]

A. Quarteroni and A. Vall, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin, 1994.  Google Scholar

[12]

A. Rashid, The pseudo-spectral collocation method for resonant long-short nonlinear wave interaction, Georgian. Math. J., 13 (2006), 143-152.   Google Scholar

[13]

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.  Google Scholar

[14]

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.   Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

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.   Google Scholar

Figure 1.  Errors for the Short wave
Figure 2.  Errors for the Long wave
Table 1.  Errors and convergence rates for the Short wave
$\tau$$L^2$-errorrate$L^\infty$-errorrate
1/105.2126e-02*2.3683e-02*
1/501.9011e-032.05747.6280e-042.1346
1/1004.7371e-042.00481.9027e-042.0033
1/5001.8917e-052.00107.6058e-062.0004
1/10004.7508e-061.99341.9052e-061.9972
$\tau$$L^2$-errorrate$L^\infty$-errorrate
1/105.2126e-02*2.3683e-02*
1/501.9011e-032.05747.6280e-042.1346
1/1004.7371e-042.00481.9027e-042.0033
1/5001.8917e-052.00107.6058e-062.0004
1/10004.7508e-061.99341.9052e-061.9972
Table 2.  Errors and convergence rates for the Long wave
$\tau$$L^2$-errorrate$L^\infty$-errorrate
1/103.7789e-02*1.5177e-02*
1/501.1696e-032.15964.7509e-042.1532
1/1002.8829e-042.02041.1614e-042.0323
1/5001.1430e-052.00554.5963e-062.0066
1/10002.8605e-061.99851.1781e-061.9640
$\tau$$L^2$-errorrate$L^\infty$-errorrate
1/103.7789e-02*1.5177e-02*
1/501.1696e-032.15964.7509e-042.1532
1/1002.8829e-042.02041.1614e-042.0323
1/5001.1430e-052.00554.5963e-062.0066
1/10002.8605e-061.99851.1781e-061.9640
Table 3.  Errors for both the Short wave and Long wave
$N$$e$$\eta$
$L^2$-error$L^{\infty}$-error$L^2$-error$L^{\infty}$-error
167.0116e-033.8134e-031.3843e-027.7596e-03
321.2206e-031.0903e-031.1102e-035.4417e-04
645.3158e-054.8213e-053.2689e-051.4387e-05
1281.2742e-064.7993e-077.3697e-073.2600e-07
$N$$e$$\eta$
$L^2$-error$L^{\infty}$-error$L^2$-error$L^{\infty}$-error
167.0116e-033.8134e-031.3843e-027.7596e-03
321.2206e-031.0903e-031.1102e-035.4417e-04
645.3158e-054.8213e-053.2689e-051.4387e-05
1281.2742e-064.7993e-077.3697e-073.2600e-07
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