Yes; the "difference" is that the "num_flux" in the two cases are different (one will contain the lapse, the other not, as undensitized) [I think]. So the implemented case with the explicit averaged lapse at each interface should directly approximate the standard form, but effectively uses a different reconstruction approach for gauge vs the rest of the flux.

I have memories of trying both approaches when playing with WENO reconstruction early on, and it making little difference. There should be no problem with scaling by any term that does not explicitly depend on terms within the hydro state vector.

From: Roland Haas <rhaas@illinois.edu>
Sent: 03 November 2021 15:31
To: Ian Hawke <I.Hawke@soton.ac.uk>
Cc: carpetx-developers@einsteintoolkit.org <carpetx-developers@einsteintoolkit.org>
Subject: Re: lapse terms in Whisky's fluxes