# American Institute of Mathematical Sciences

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:

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##### References:
Errors for the Short wave
Errors for the Long wave
Errors and convergence rates for the Short wave
 $\tau$ $L^2$-error rate $L^\infty$-error rate 1/10 5.2126e-02 * 2.3683e-02 * 1/50 1.9011e-03 2.0574 7.6280e-04 2.1346 1/100 4.7371e-04 2.0048 1.9027e-04 2.0033 1/500 1.8917e-05 2.0010 7.6058e-06 2.0004 1/1000 4.7508e-06 1.9934 1.9052e-06 1.9972
 $\tau$ $L^2$-error rate $L^\infty$-error rate 1/10 5.2126e-02 * 2.3683e-02 * 1/50 1.9011e-03 2.0574 7.6280e-04 2.1346 1/100 4.7371e-04 2.0048 1.9027e-04 2.0033 1/500 1.8917e-05 2.0010 7.6058e-06 2.0004 1/1000 4.7508e-06 1.9934 1.9052e-06 1.9972
Errors and convergence rates for the Long wave
 $\tau$ $L^2$-error rate $L^\infty$-error rate 1/10 3.7789e-02 * 1.5177e-02 * 1/50 1.1696e-03 2.1596 4.7509e-04 2.1532 1/100 2.8829e-04 2.0204 1.1614e-04 2.0323 1/500 1.1430e-05 2.0055 4.5963e-06 2.0066 1/1000 2.8605e-06 1.9985 1.1781e-06 1.9640
 $\tau$ $L^2$-error rate $L^\infty$-error rate 1/10 3.7789e-02 * 1.5177e-02 * 1/50 1.1696e-03 2.1596 4.7509e-04 2.1532 1/100 2.8829e-04 2.0204 1.1614e-04 2.0323 1/500 1.1430e-05 2.0055 4.5963e-06 2.0066 1/1000 2.8605e-06 1.9985 1.1781e-06 1.9640
Errors for both the Short wave and Long wave
 $N$ $e$ $\eta$ $L^2$-error $L^{\infty}$-error $L^2$-error $L^{\infty}$-error 16 7.0116e-03 3.8134e-03 1.3843e-02 7.7596e-03 32 1.2206e-03 1.0903e-03 1.1102e-03 5.4417e-04 64 5.3158e-05 4.8213e-05 3.2689e-05 1.4387e-05 128 1.2742e-06 4.7993e-07 7.3697e-07 3.2600e-07
 $N$ $e$ $\eta$ $L^2$-error $L^{\infty}$-error $L^2$-error $L^{\infty}$-error 16 7.0116e-03 3.8134e-03 1.3843e-02 7.7596e-03 32 1.2206e-03 1.0903e-03 1.1102e-03 5.4417e-04 64 5.3158e-05 4.8213e-05 3.2689e-05 1.4387e-05 128 1.2742e-06 4.7993e-07 7.3697e-07 3.2600e-07
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