Hesthaven's book 1 mentions the following time step restriction for Navier-Stokes equations (see (7.32) in 2008 edition) $$ \Delta t \approx \frac{h}{N^2} \frac{C}{|u| + |c| + \frac {N^2 \mu}{h}} $$ ($h$-element size, $N$-degree of polynomial basis, $C$-constant)

Isn't this equation dimensionally incorrect? $\mu/h$ doesn't have dimensions of $u$. I think $\mu$ should be replaced by the kinematic viscosity $\nu$. However, there is nothing mentioned about this in "corrections" either (2,3).

Can anyone help me with giving the correct form of this?


$\Delta t$ - time step

$h$ - characteristic element size (usually radius of inscribed or circumscribed circle)

$N$ - degree of polynomial approximation

$C$ - constant, $O(1)$

$u$ - fluid velocity in $x$ direction

$c$ - speed of sound

$\mu$ - dynamic viscosity

$\nu$ - kinematic viscosity

  • $\begingroup$ In my experience a lot of times dimensionful factors are used, and if they are set to 1 they are left out completely. Maybe this is one of those cases? $\endgroup$
    – Emil
    Oct 28 '20 at 7:12
  • $\begingroup$ I think that might not be the case here. Time step restriction for a purely diffusive problem is $O(h^2/\nu)$. So I feel having $\nu$ in the expression is more natural. $\endgroup$
    – Zxcvasdf
    Oct 28 '20 at 9:34
  • $\begingroup$ @DavidKetcheson I have edited my post to include description of the symbols $\endgroup$
    – Zxcvasdf
    Oct 30 '20 at 6:10
  • $\begingroup$ I haven't checked everything, but I think you're missing the fact that $N$ effectively has dimensions of 1/length, at least for the purposes of a time step condition. $\endgroup$ Oct 30 '20 at 7:53
  • $\begingroup$ @DavidKetcheson I don't think so. Here $N$ is just the degree of polynomial approximation, a number. For instance, in pure advection flows, it is a common practice in DG methodology to divide the finite volume stable time step by $(2N+1)$ to get the corresponding DG value. $\endgroup$
    – Zxcvasdf
    Oct 30 '20 at 12:35

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