Discrepancy between Navier-Stokes into and first example

[rehwinkel]

In the Navier-Stokes introductory page, an intermediate step is described here. By the definition of $u^{n+\frac12} = \frac12 (u^{n}+u^{n+1})$, the equation depends on $u^\ast$ as well as $u^{n+1}$. In the first example problem, it seems $u^\ast$ is implicitly assumed to be the same as $u^{n+1}$:

# previously
U = 0.5 * (u_n + u)
# ...
p_n = Function(Q)
p_n.name = "p_n"
F1 = rho * dot((u - u_n) / k, v) * dx # here
F1 += rho * dot(dot(u_n, nabla_grad(u_n)), v) * dx
F1 += inner(sigma(U, p_n), epsilon(v)) * dx
F1 += dot(p_n * n, v) * ds - dot(mu * nabla_grad(U) * n, v) * ds
F1 -= dot(f, v) * dx
a1 = form(lhs(F1))
L1 = form(rhs(F1))

I believe this discrepancy is resolved by instead defining:
$u^{n+\frac12} := \frac12 (u^{n}+u^{*})$

[jorgensd]

Yes, agreed.
I’ve given a more descriptive explaination and derivation at: https://computationalphysiology.github.io/oasisx/splitting_schemes.html#navier-stokes-equation

If you want to make a PR that corrects the ipynb and py code (you can update one, and use jupytertext to convert it to the other, see the readme), I can review and merge it.