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 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 :
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 :
So, I tried another possibility to code this term :