A parametric evaluation and grid-resolution study of the Korteweg–de Vries equation using the Fourier pseudospectral method
DOI:
https://doi.org/10.55573/JISED.118765Keywords:
Korteweg–de Vries equation, Fourier pseudospectral method, grid resolution, dealiasing, soliton fission, dispersive wavesAbstract
The Fourier pseudospectral method is a well-established numerical approach for solving partial differential equations (PDEs). One of the most well-known PDEs is the classical Korteweg–de Vries (KdV) equation which models shallow-water wave systems such as tsunamis and tidal bores. The qualitative behaviour of KdV solutions changes accordingly with the relative strengths of nonlinearity and dispersion, and reliable numerical capture of these different regimes depends on spatial resolution, aliasing control, and computational-domain selection. However, these numerical issues are often considered separately rather than systematically within a single Fourier pseudospectral framework. Here, the Fourier pseudospectral method is employed to study the nonlinear dispersive wave propagation. Spatial derivatives are evaluated in Fourier space, while the nonlinear terms are computed in physical space using the standard 2/3 dealiasing rule to reduce aliasing errors. Since the accuracy of this method depends strongly on the spatial resolution, a grid-resolution study is first conducted using different numbers of spatial grid points, N, to determine their influence on the numerical solution. Three parameter regimes are considered to examine the effects of the relative strengths of nonlinearity and dispersion. In the balanced regime, the amplitude and the phase speed of the solitary wave is maintained providing a validation of the numerical implementation and parameter selection. In the nonlinearity-dominated regime, increasing the nonlinear coefficient produces wave steepening followed by soliton fission. Conversely, increasing the dispersion coefficient leads to amplitude reduction, dispersive spreading, and the development of oscillatory wave trains. The results demonstrate that the Fourier pseudospectral method with appropriate spatial resolution and 2/3 dealiasing can accurately resolve qualitatively different KdV wave regimes when computational-domain size is appropriately selected.
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