How does the FOVE system work with the Airbus FMS?
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Whether the FMS is capable or not of calculating the V-speeds, it should be checked and may be overridden. It is because the airplane database wouldn't know if there's a temporary crane that will affect the climb limit weight, or if 400 ft of the runway are unusable due to maintenance, for example. In other words, 10,000' corrected-length runways are not always the same.

As to FOVE, it is an Airbus tool made by Airbus. The Airbus LPC (Less Paper in the Cockpit) is a laptop* used by the pilots. Essentially an EFB (electronic flight bag). This laptop serves two functions. One, is having the manuals in electronic format, two, is the FOVE—which handles all the performance related calculations.

* Depending on the company, the laptop can be a tablet, or an integrated EFB. I can't speak for the integrated, but the laptop/tablet do not communicate with the FMS, and changes are done manually.

It is more accurate than manual calculation, because manual calculations rely on approved graphs. Interpreting graphs inherently reduces accuracy. See below for a 737-800 example:

(Click to view)

And that's for one flap setting and one condition.

Even if converted to table format, still there will be gaps between any two values, whereas a computer can produce those 'in-between' values, which may provide a takeoff performance advantage.

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See here for FOVE training manual that has lots of screenshots, etc.