# Explicit DG time step restriction for compressible Navier-Stokes equations

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?

## Edit

$$\Delta t$$ - 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

• 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?
– Emil
Commented Oct 28, 2020 at 7:12
• 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. Commented Oct 28, 2020 at 9:34
• @DavidKetcheson I have edited my post to include description of the symbols Commented Oct 30, 2020 at 6:10
• 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. Commented Oct 30, 2020 at 7:53
• @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. Commented Oct 30, 2020 at 12:35