We now study the Kuramoto–Sivashinsky equation,

\[\pd{u}{t} = -\nabla^2u-\nabla^4u-|\nabla u|^2,\]

with periodic boundary conditions.

  • Load the interactive simulation. If you perturb the solution, it should devolve into a kind of spatiotemporal chaos of oscillation and movement. Importantly, the patterns which emerge have a certain set of coherent wavelengths, which suggests that the dynamics is that of finite-dimensional chaos, rather than fully turbulent mixing.

Numerical notes

The equation above is far from the cross-diffusion kind of system our solver is built for. However, using the product rule and an algebraic substitution, we can write it as:

\[\begin{align} \pd{u}{t}& = -\vnabla \cdot [ (1+u)\vnabla u + \vnabla v]+uv-au, \\ v& = \nabla^2 u, \end{align}\]

where $a$ is a damping coefficient used to help stabilise the solver. For $a=0$, this is exactly the fourth-order equation above.