Concerning the equation

The KSE is a partial differential equation. One of its terms is not linear and another is a fourst order spatial derivation. This observations imply that we need $5$ points to approximate the higher spatial partial derivation, and that we have to take great care of the modelisation of the unlinear term.

As the flow can have a positive or a negative velocity, I have to take a center scheme to approximate the spatial derivation. It is impossible to privilege a direction. As the explicit center schemes are unstable, I have been obliged to use an implicit center scheme.

The time discretisation has been realized with the classical expression :

$$\left. \frac{\partial u}{\partial t}\right\vert _i^n=\frac{u_i^{n+1}-u_i^n} {\Delta t}$$

For the modelisation of the unlinear term of the KSE there is a natural choise. It will be modelised with an explicit scheme because it is the easiest solution.

I'd like make a very considerable remark on the modelisation of the unlinear term of the KSE. I had decided first to code this term is this way :

$$u\frac{\partial u}{\partial x}=u_i^n \frac{u_{i+1}^{n}-u_{i-1}^n}{2 \Delta x}$$

I had observed that this modelisation wasn't good enough because, I didn't find again the results describe in [1]

So, I tried another possibility to code this term :

$$u\frac{\partial u}{\partial x}=\frac{{u_{i+1}^{n}}^2-{u_{i-1}^n}^2} {2 \Delta x}$$

This possibility gives no more results. The course ``turbulence simulation'' gave me the solution. To simulate correctly the turbulence, the code must uses a modelisation of the unlinear term which is the less dissipative as possible. This modelisation is :

$$u\frac{\partial u}{\partial x}=\frac{1}{2}\left( u_i^n \frac{u_{i... ...{i-1}^n}{2 \Delta x} +\frac{{u_{i+1}^{n}}^2-{u_{i-1}^n}^2}{4 \Delta x} \right)$$

With this discretisation, I have been able to make the study of the bifurcation of the KSE.