Kuramoto-Sivashinsky Module
Basic Equations
This module solves the Kuramoto-Sivashinsky equation for $u(x,t)$:
\[\partial_t u + \partial_x^4 u + \partial_x^2 u + u\partial_x u = 0\ .\]
Implementation
The equation is time-stepped forward in Fourier space:
\[\partial_t \widehat{u} + k_x^4 \widehat{u} - k_x^2 \widehat{u} + \widehat{ u\partial_x u } = 0\ .\]
Thus:
\[\mathcal{L} = k_x^2 - k_x^4\ ,\]
\[\mathcal{N}(\widehat{u}) = - \mathrm{FFT}(u \partial_x u)\ .\]
The function calcN! implements dealiasing to avoid energy piling up at the grid-scale.