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Volume 37, Issue 5
Efficient and Accurate Numerical Methods for Long-Wave Short-Wave Interaction Equations in the Semiclassical Limit Regime

Tingchun Wang, Xiaofei Zhao, Mao Peng & Peng Wang

DOI: 10.4208/jcm.1807-m2015-0271

J. Comp. Math., 37 (2019), pp. 645-665.

Published online: 2019-03

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  • Abstract

This paper focuses on performance of several efficient and accurate numerical methods for the long-wave short-wave interaction equations in the semiclassical limit regime. The key features of the proposed methods are based on: (i) the utilization of the first-order or second-order time-splitting method to the nonlinear wave interaction equations; (ii) the application of Fourier pseudo-spectral method or compact finite difference approximation to the linear subproblem and the spatial derivatives; (iii) the adoption of the exact integration of the nonlinear subproblems and the ordinary differential equations in the phase space. The numerical methods under study are efficient, unconditionally stable and higher-order accurate, they are proved to preserve two invariants including the position density in $L^1$. Numerical results are reported for case studies with different types of initial data, these results verify the conservation laws in the discrete sense, show the dependence of the numerical solution on the time-step, mesh-size and dispersion parameter ε, and demonstrate the behavior of nonlinear dispersive waves in the semi-classical limit regime.

  • Keywords

Long-wave short-wave interaction equations, Semiclassical limit, Time-splitting method, Spectral method, Compact finite difference method, Conservative properties.

  • AMS Subject Headings

65M06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

wangtingchun2010@gmail.com (Tingchun Wang)

zhxfnus@gmail.com (Xiaofei Zhao)

mpengmath@163.com (Mao Peng)

pwang@jlu.edu.cn (Peng Wang)

  • BibTex
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  • TXT
@Article{JCM-37-645, author = {Wang , TingchunZhao , XiaofeiPeng , Mao and Wang , Peng}, title = {Efficient and Accurate Numerical Methods for Long-Wave Short-Wave Interaction Equations in the Semiclassical Limit Regime}, journal = {Journal of Computational Mathematics}, year = {2019}, volume = {37}, number = {5}, pages = {645--665}, abstract = {

This paper focuses on performance of several efficient and accurate numerical methods for the long-wave short-wave interaction equations in the semiclassical limit regime. The key features of the proposed methods are based on: (i) the utilization of the first-order or second-order time-splitting method to the nonlinear wave interaction equations; (ii) the application of Fourier pseudo-spectral method or compact finite difference approximation to the linear subproblem and the spatial derivatives; (iii) the adoption of the exact integration of the nonlinear subproblems and the ordinary differential equations in the phase space. The numerical methods under study are efficient, unconditionally stable and higher-order accurate, they are proved to preserve two invariants including the position density in $L^1$. Numerical results are reported for case studies with different types of initial data, these results verify the conservation laws in the discrete sense, show the dependence of the numerical solution on the time-step, mesh-size and dispersion parameter ε, and demonstrate the behavior of nonlinear dispersive waves in the semi-classical limit regime.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1807-m2015-0271}, url = {http://global-sci.org/intro/article_detail/jcm/13039.html} }
TY - JOUR T1 - Efficient and Accurate Numerical Methods for Long-Wave Short-Wave Interaction Equations in the Semiclassical Limit Regime AU - Wang , Tingchun AU - Zhao , Xiaofei AU - Peng , Mao AU - Wang , Peng JO - Journal of Computational Mathematics VL - 5 SP - 645 EP - 665 PY - 2019 DA - 2019/03 SN - 37 DO - http://doi.org/10.4208/jcm.1807-m2015-0271 UR - https://global-sci.org/intro/article_detail/jcm/13039.html KW - Long-wave short-wave interaction equations, Semiclassical limit, Time-splitting method, Spectral method, Compact finite difference method, Conservative properties. AB -

This paper focuses on performance of several efficient and accurate numerical methods for the long-wave short-wave interaction equations in the semiclassical limit regime. The key features of the proposed methods are based on: (i) the utilization of the first-order or second-order time-splitting method to the nonlinear wave interaction equations; (ii) the application of Fourier pseudo-spectral method or compact finite difference approximation to the linear subproblem and the spatial derivatives; (iii) the adoption of the exact integration of the nonlinear subproblems and the ordinary differential equations in the phase space. The numerical methods under study are efficient, unconditionally stable and higher-order accurate, they are proved to preserve two invariants including the position density in $L^1$. Numerical results are reported for case studies with different types of initial data, these results verify the conservation laws in the discrete sense, show the dependence of the numerical solution on the time-step, mesh-size and dispersion parameter ε, and demonstrate the behavior of nonlinear dispersive waves in the semi-classical limit regime.

Tingchun Wang, Xiaofei Zhao, Mao Peng & Peng Wang. (2019). Efficient and Accurate Numerical Methods for Long-Wave Short-Wave Interaction Equations in the Semiclassical Limit Regime. Journal of Computational Mathematics. 37 (5). 645-665. doi:10.4208/jcm.1807-m2015-0271
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