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Abstract
In the field of water waves, the modified nonlinear Schrödinger (mNLS) equation which models the wave propagation in water is numerically solved by using the split-step pseudo-spectral method. In the present paper, the fourth-order split-step pseudo-spectral method is introduced with better numerical results. The proposed method is based on a split-step method which decomposes the original equation into two parts, a linear problem and a nonlinear problem. In order to demonstrate the high accuracy and capability of the newly proposed method, a simple problem of periodic waves is presented to compare the traditional first-order method with the fourth-order method in terms of the conservation error and computational cost. Meanwhile, another numerical experiment concerning the Peregrine breather solution of the nonlinear Schrödinger (NLS) equation is presented by using the fourth-order split-step pseudo-spectral method. It is found that the fourth-order scheme can provide higher conservative accuracy and is computationally more efficient compared with the first-order scheme.
Acknowledgements
This work was financially supported by the National Natural Science Foundation of China (Grant No. 51239007). The authors would also like to acknowledge the support of the Sino-UK Higher Education Research Partnership for PhD Studies funded by the British Council in China and the China Scholarship Council.
Disclosure statement
No potential conflict of interest was reported by the authors.
| Nomenclature | ||
| ϕ | = | Velocity potential (m2/s)] |
| ζ | = | Wave surface elevation (m) |
| ϕ | = | Velocity potential of mean flow (m2/s) |
| ζ | = | Wave surface elevation of mean flow (m) |
| A,A2,A3 | = | Complex displacement amplitudes (m) |
| B,B2,B3 | = | Complex velocity potential amplitudes (m2/s) |
| k | = | Wave number (m−1) |
| ω | = | Wave frequency (s−1) |
| cg | = | Wave group velocity (m/s) |
| x | = | Real space variable (m) |
| t | = | Real time variable (s) |
| a | = | Wave amplitude (m) |
| γ | = | Scale factor |
| ϵ | = | Wave steepness |
| ξ | = | Dimensionless time variable |
| η | = | Dimensionless space variable |
| Ψ | = | Phase function |
| υ | = | Fourier mode |
| q | = | Dimensionless complex wave amplitude |
| X | = | Dimensionless time variable |
| T | = | Dimensionless space variable |
| Ω, p, β | = | Intermediate variables |