One dimensional Kuramoto-Sivashinsky ETDRK4 solver

This code solves the one-dimensional Kuramoto-Sivashisnky equation with periodic boundary condition

$$\left\{ \begin{array}{ll} \displaystyle \frac{\partial u}{\partial t} + \frac{\partial^{2} u}{\partial x^{2}} + \nu \frac{\partial^{4} u}{\partial x^{4}} + \frac{1}{2} \frac{\partial u^{2}}{\partial x} = 0 \\\ u(x+L,t) = u(x,t) \\\ x \in [0,L], L \in \mathbb{R}_{+}^{*} \\\ \nu \in ]0,1[ \\\ t \in \mathbb{R}_{+} \end{array} \right.$$

using spectral method with an ETDRK4 solver (as the equation is stiff).

More precisly, since the solution u is $L$-periodic, we decompose it into a Fourier basis

$$u(x,t) = \sum_{k \in \mathbb{Z}} \hat u_{k}(t)e^{i\frac{2\pi}{L}kx}$$

which leads to the following ODE

$$\frac{d \hat u_{k}}{dt} = \left(q_{k}^{2} - q_{k}^{4}\right) \hat u_{k} - \frac{iq_{k}}{2} \mathcal{F} \Bigl[ \left(\mathcal{F}^{-1}[\hat u] \right)^{2} \Bigr]_{k}$$

where $\displaystyle q_{k} = \left(\frac{2\pi}{L}\right)k$ and $\displaystyle \mathcal{F}$ is the Fourier transform.
The ODE is solved using an ETDRK4, which is particularly well suited to stiff equations.

Moreover, the associated linear equation is also solved in tangent space, which allows to compute Lyapunov exponents and Covariant Lyapunov Vectors (CLV). An example is given in the notebook.

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If you find any error or bug in the code, do not hesitate to create an issue or a pull request.