Implicit, high-order methods for Incompressible NS

Dave Rudolf

Hey all,

I am just curious as to how one would develop an implicit solver for the incompressible Navier-Stokes equations, with finite differences to handle the spatial derivatives.

I am quite familiar with methods for ODEs (like backward Euler, Adams-Bashford/Adams-Moulton, BDFs, RKs, etc.). I have also seen the usual explicit Eulerian approach to solving NS, which differs from how one would solve an ODE. I am talking about the explicit method presented by Griebel, Dornseifer, Neunhoeffer in their book "Numerical Simulation in Fluid Dynamics". They way they describe it, one would take a forward-Euler step, and then solve the Poisson equation for pressure to make the system divergence-free.

I can't seem to find any decent explanations of how NS is solved with implicit methods, or with higher-order methods. I can think of two ways, but haven't been able to find info on either one of them. Either there is a way to reduce NS to an ODE, or there are special methods (like the explicit one by Griebel et al.) that only work with NS.

So, am I missing something? Can someone point me to some info?

Thanks.

Dave