# Adding an external force to Chorin's projection method for the Navier-Stokes equation

I am reading the paper, http://math.mit.edu/~gs/cse/codes/mit18086_navierstokes.pdf

I want to solve an optimization problem using this Navier-Stokes code as constraining PDE. Now I am trying to add an external force parameterized some input parameter to the method. The method solves the Navier-Stokes equation in 3 steps, 1) nonlinear term, 2) linear term, 3) pressure correction. $$\text{(1)} \quad \frac{U^*-U^n}{\Delta t} = -((U^n)^2)_x - (U^nV^n)_y$$ $$\text{(2)} \quad \frac{U^**-U^*}{\Delta t} = \frac{1}{Re}(U^{**}_{xx}+U^{**}_{yy})$$ $$\text{(3)} \quad U{n+1} = U^{**} - \Delta t \nabla P^{n+1}$$ (written only for U)

Should I add the external force term to the step 1 like this? $$\text{(1)} \quad \frac{U^*-U^n}{\Delta t} = -((U^n)^2)_x - (U^nV^n)_y +f$$

But then, will the external force corresponds to the input be modified to satisfy the incompressible condition at step 3?

If I give some physically impossible external force term (like too much force), the method will automatically decrease it to satisfy the incompressible condition?

## 1 Answer

I would not bother too much. Chorin's method is known to be a consistent scheme, which means, that for smaller time-steps it will approximate the actual solution. Checkout Gresho's/Sani's book Incompressible Flow and the Finite Element Method. Vol. 2: Isothermal Laminar Flow there is the convergence theory and also Chorin's method with a forcing term.

So, just put your force into $(1)$ or $(2)$ -- there is no difference.

The last step $(3)$ is a projection step that projects your tentative velocity $U^{**}$ to a divergence-free corrected velocity. It will remove certain parts of $f$, namely those that entered the gradient-part of $U^{**}$. In fact, if $f=\nabla \rho$, where $\rho$ is suitable scalar function, then (in theory) $f$ should never appear in the velocity and then, at best, also not in $U^{**}$.

So, to address your last points... It's not about too much force, it is about the direction of the force (whether it is a gradient for example). And Chorin's method will project out the gradient parts up to the consistency error.