Time discretization Navier Stokes equation

This question is a follow-up of this one.

The weak form of Navier Stokes equation is (assuming $$v,q$$ test functions for the velocity and the pressure, respectively)

$$(\frac{du}{dt},v)_{\Omega} + (\nabla u, \nabla v) _{\Omega}- (\operatorname{div}(v),p)_{\Omega} - (q,\operatorname{div}(u))_{\Omega} + c(u,u,v)_{\Omega}= (v,f)_{\Omega}$$

Now I've been told to advance in time using a time integrator (so I really need to use some available routine like the ones provided by SUNDIALS, instead of doing the time-stepping on my own). The system above becomes the following one:

$$M \dot{u} = f- Au -B^tp - C(u(t))u(t)$$ $$B u(t)=0$$

but I really do not see the usual form of an ODE. (I was thinking to a DAE, but my professor explicitely told us that we don't have to solve a DAE). I'm using deal.II, so I need to use the SUNDIALS wrappers to advance in time. My biggest problem is that I can't "see" that system as a system of ODEs. What am I missing?

• If you don't want to solve this as ODE+DAE, then the only other option is to use fractional-step (or projection) methods. See this tutorial from deal.ii. dealii.org/current/doxygen/deal.II/step_35.html Commented Jul 5, 2021 at 10:12
• If you make that run with SUNDIALS, it would be very nice to eventually get that program into the deal.II code gallery! Commented Jul 5, 2021 at 16:57
• @WolfgangBangerth I'd be happy to share it, of course. The only thing I need to clarify is how I can use arkode interface. I mean, $$M \dot{u} = f- Au -B^tp - C(u(t))u(t)$$ is certainly in the right form, but with the incompressibility constraint I don't have an ODE anymore. Commented Jul 5, 2021 at 18:36
• The N-S equations are definitely a DAE, but I can't say whether one would solve it as a DAE when using SUNDIALS. The methods I'm familiar with are of projection type, and they generally are built using hand-written time integration loops. But that may not be the most efficient way. If you have the time, I would just give it a try with SUNDIALS' IDA integrator and if that doesn't work, try a different approach. Commented Jul 5, 2021 at 22:19
• Also, ask your professor why he thinks that you shouldn't see the equation as a DAE -- teaching is one of the jobs of professors :-) Commented Jul 5, 2021 at 22:19

I don't see what you're not seeing. You've written out a perfectly good non-linear weak form of the PDE in your last two equations. If you invert $$M$$ and apply it, then you have a non-linear equation for $$\dot{u}(t)$$ that you can try to solve. It's a continuous in space and time PDE that you would need to discretize in space to then have only an ODE, with $$M$$, $$C$$, $$A$$, and $$B$$ being space/time operators that need spatial discretization in space by introducing some finite element grid/mesh functions (e.g., $$u_h(t)$$) to generate a method that is only an ODE in time. Also, you need some sort of initial conditions (ICs) for $$u_(t_0)$$. You also need boundary conditions on $$\Omega$$ for $$u_h(t)$$. You also need to linearize $$c$$ so that you end up with a solvable system. My notation is a little loose here, but I think all the ideas are there.
• The fact is that I need to advance in time using SUNDIALS wrappers (dealii.org/current/doxygen/deal.II/classSUNDIALS_1_1ARKode.html). Yes, the first equation is an ODE, while the second contraint $Bu=0$ makes the system a DAE. (here with you $u$ I actually means $u_h$, I already applied a FE discretization in the last equations). Therefore I cannot see how I can use that interface. Commented Jul 5, 2021 at 17:49
• @Vefhug, it looks like to me everything you need is in the deal.ii tutorials and lectures. Professor Bangerth's videos about this are on YouTube as well. In your case $M$ is $M$ (i.e. the mass matrix) and you're going to have to figure out your Jacobian, but the pieces are basically there. Commented Jul 5, 2021 at 18:46
• So, are you going to solve only $$M \dot{u} = f- Au -B^tp - C(u(t))u(t)$$ ? I'm really sorry to keep asking, but I looked up all the deal.ii tutorials in the past days, and they never use an ODE solver to solve the time dependent NS equation. Commented Jul 5, 2021 at 19:45