Hermite spectral method for Long-Short wave equations

Shujuan Lü; Zeting Liu; Zhaosheng Feng

doi:10.3934/dcdsb.2018255

Article Contents

2019, Volume 24, Issue 2: 941-964. Doi: 10.3934/dcdsb.2018255

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

Received: December 31, 2015

Revised: June 30, 2018

Early access: October 2018

Published: January 2019

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Abstract

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.

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

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Figure 1. Errors for the Short wave

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Figure 2. Errors for the Long wave

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Table 1. 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

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Table 2. 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

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

$N$	$e$		$\eta$
$N$	$L^2$-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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