I don't know why we have this in the first place, but the particle computation is shifted at one place. The assumption about the velocity to be present at time step u_n+1 makes no sense.
I therefore have to either shift all computations of the velocity by one timestep (which makes them again way more expensive) or introduce a shifted computation for the particles alltogether. Since they are kinda decoupled, maybe this is a clever way to deal with this, maintain the nice speed properties and not having to change all present schemes and do the convergence tests all over again.
This should be discussed with Kai and Thibault.
I don't know why we have this in the first place, but the particle computation is shifted at one place. The assumption about the velocity to be present at time step u_n+1 makes no sense. I therefore have to either shift all computations of the velocity by one timestep (which makes them again way more expensive) or introduce a shifted computation for the particles alltogether. Since they are kinda decoupled, maybe this is a clever way to deal with this, maintain the nice speed properties and not having to change all present schemes and do the convergence tests all over again. This should be discussed with Kai and Thibault.