Numerical Simulation of the Solitary Wave Collision Process in Time Fractional Orders Based on the Coupled Pure Meshless Method

LI Yue; JIANG Rongrong; JIANG Tao

doi:10.21656/1000-0887.420278

Volume 43 Issue 9

Sep. 2022

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Article Navigation > Applied Mathematics and Mechanics > 2022 > 43(9): 1016-1025

LI Yue, JIANG Rongrong, JIANG Tao. Numerical Simulation of the Solitary Wave Collision Process in Time Fractional Orders Based on the Coupled Pure Meshless Method[J]. Applied Mathematics and Mechanics, 2022, 43(9): 1016-1025. doi: 10.21656/1000-0887.420278

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Numerical Simulation of the Solitary Wave Collision Process in Time Fractional Orders Based on the Coupled Pure Meshless Method

doi: 10.21656/1000-0887.420278

College of Mathematical Science, Yangzhou University, Yangzhou, Jiangsu 225002, P.R.China

Received Date: 2021-09-13
Rev Recd Date: 2021-11-12

Available Online: 2021-09-29

Publish Date: 2022-09-30

Abstract

Abstract

A coupled pure meshless finite pointset method (CFPM) was developed for the first time to numerically predict the inelastic collision process of solitary waves described with the time fractional coupled nonlinear Schrödinger (TF-CNLS) equation. Its construction process was formulated as: 1) a high-precision difference scheme was used for the Caputo time fractional derivative; 2) the FPM discrete scheme based on the Taylor expansion and the weighted least square method was adopted for spatial derivatives; 3) the region was locally refined and the double cosine kernel function with good stability was used to improve the numerical accuracy. In the numerical study, the 1D TF-CNLS equations were analytically solved with the CFPM, and the errors and convergence rates were analyzed with the nodes uniformly distributed or locally refined, which shows that the proposed method has the approximate 2nd-order accuracy and the flexibility of easy local refinement. Secondly, the inelastic collision process of solitary waves, which was described with the 1D TF-CNLS equation without analytical solutions, was numerically predicted with the CFPM, and the wave collapse phenomenon is completely different from the multi-wave phenomenon in the integer order. Meanwhile, the comparison of the results with those from the finite difference method shows that, the CFPM is reliable to predict the complex propagation of the inelastic collision process of the solitary waves in the time fractional order.
- Caputo fractional derivative,
- difference method,
- FPM,
- TF-CNLS,
- solitary wave collision

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References

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