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Volume 31, Issue 4
Numerical Integrators for Dispersion-Managed KdV Equation

Ying He & Xiaofei Zhao

DOI: 10.4208/cicp.OA-2021-0216

Commun. Comput. Phys., 31 (2022), pp. 1180-1214.

Published online: 2022-03

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  • Abstract

In this paper, we consider the numerics of the dispersion-managed Korteweg-de Vries (DM-KdV) equation for describing wave propagations in inhomogeneous media. The DM-KdV equation contains a variable dispersion map with discontinuity, which makes the solution non-smooth in time. We formally analyze the convergence order reduction problems of some popular numerical methods including finite difference and time-splitting for solving the DM-KdV equation, where a necessary constraint on the time step has been identified. Then, two exponential-type dispersion-map integrators up to second order accuracy are derived, which are efficiently incorporated with the Fourier pseudospectral discretization in space, and they can converge regardless of the discontinuity and the step size. Numerical comparisons show the advantage of the proposed methods with the application to solitary wave dynamics and extension to the fast & strong dispersion-management regime.

  • Keywords

KdV equation, dispersion management, discontinuous coefficient, convergence order, finite difference, time-splitting, exponential integrator, pseudospectral method.

  • AMS Subject Headings

65M12, 65M15, 65M70, 65Z05, 35Q53

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-31-1180, author = {He , Ying and Zhao , Xiaofei}, title = {Numerical Integrators for Dispersion-Managed KdV Equation}, journal = {Communications in Computational Physics}, year = {2022}, volume = {31}, number = {4}, pages = {1180--1214}, abstract = {

In this paper, we consider the numerics of the dispersion-managed Korteweg-de Vries (DM-KdV) equation for describing wave propagations in inhomogeneous media. The DM-KdV equation contains a variable dispersion map with discontinuity, which makes the solution non-smooth in time. We formally analyze the convergence order reduction problems of some popular numerical methods including finite difference and time-splitting for solving the DM-KdV equation, where a necessary constraint on the time step has been identified. Then, two exponential-type dispersion-map integrators up to second order accuracy are derived, which are efficiently incorporated with the Fourier pseudospectral discretization in space, and they can converge regardless of the discontinuity and the step size. Numerical comparisons show the advantage of the proposed methods with the application to solitary wave dynamics and extension to the fast & strong dispersion-management regime.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0216}, url = {http://global-sci.org/intro/article_detail/cicp/20381.html} }
TY - JOUR T1 - Numerical Integrators for Dispersion-Managed KdV Equation AU - He , Ying AU - Zhao , Xiaofei JO - Communications in Computational Physics VL - 4 SP - 1180 EP - 1214 PY - 2022 DA - 2022/03 SN - 31 DO - http://doi.org/10.4208/cicp.OA-2021-0216 UR - https://global-sci.org/intro/article_detail/cicp/20381.html KW - KdV equation, dispersion management, discontinuous coefficient, convergence order, finite difference, time-splitting, exponential integrator, pseudospectral method. AB -

In this paper, we consider the numerics of the dispersion-managed Korteweg-de Vries (DM-KdV) equation for describing wave propagations in inhomogeneous media. The DM-KdV equation contains a variable dispersion map with discontinuity, which makes the solution non-smooth in time. We formally analyze the convergence order reduction problems of some popular numerical methods including finite difference and time-splitting for solving the DM-KdV equation, where a necessary constraint on the time step has been identified. Then, two exponential-type dispersion-map integrators up to second order accuracy are derived, which are efficiently incorporated with the Fourier pseudospectral discretization in space, and they can converge regardless of the discontinuity and the step size. Numerical comparisons show the advantage of the proposed methods with the application to solitary wave dynamics and extension to the fast & strong dispersion-management regime.

He , Ying and Zhao , Xiaofei. (2022). Numerical Integrators for Dispersion-Managed KdV Equation. Communications in Computational Physics. 31 (4). 1180-1214. doi:10.4208/cicp.OA-2021-0216
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