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Usually I rappel using a tubular belay device such as an ATC and tie a stopper knot at the end of each strand of the rope such as a fisherman’s knot.
Is there a safety knot which you can tie before rappelling using the Munter hitch (instead of an ATC) on a double-stranded rope which does not pass through your carabiner?
TLDR; Join yourself to the ends of the rope.
My preference if the rope isn't on the ground is towards tieing an overhand on a bight on each side and clipping them both my belay loop.
This strategy has two benefits major benefits.
1) The obvious is that when rappelling, you can't go past the end of the rope. This is true regardless of which type of friction device you are using.
2) The second is less obvious, but self-rescue becomes a lot easier if you can easily remove the device and still be attached to the system.
Noteworthy: I have seen people tie a giant knot (BHK/BFK), but I think you lose versatility when doing so. That said, it's not likely to pass through your munter hitch (depending on the diameter of rope).
The Prusik: The prusik hitch doesn't prevent you from falling off the end, and in some cases is dangerous (see: Canyoning; rappelling down waterfalls specifically). If you use one though, any stopper/catastrophe knot will prevent the prusik from sliding past it.
It's good that you're in the habit of putting knots in the end of your rope before you rappel. Failing to do that is an extremely common cause of accidents and deaths, because without the knots, you can pop off the end of the rope unexpectedly (or pop off of one strand, if they're uneven).
Doing this with a Munter is not 100% guaranteed to work, because it is physically possible to pass a knot through a Munter, if the Munter is loose. This is different from an ATC, where it's simply not possible for the knot to pass through the device. However, the reality is that a stopper knot is very likely to jam and not pass through the Munter, if the rope under tension.
Also, if you reach the knotted end of the rope, you will feel the knot come up against your braking hand, so you have some warning. For this reason, it should never even be an issue whether the knot could somehow slip through the Munter. Even in a normal rappel with an ATC, we'd prefer not to let the stopper knot reach the ATC. If that happens, it can get jammed, and you can have problems with escaping from the situation by ascending. The reason people learn techniques like the Munter mule overhand is that you want to be able to stop securely without letting a knot jam into your belay device.
In some of the other answers and comments, it seems like people are misreading the question or not understanding the distinction between two different types of safety backups for rappelling. You want to guard against two types of errors: (1) losing control of the rappel and hitting the ground or a ledge; (2) rappelling off the end of the rope. The question is clearly about #2. Safety measures like a Prusik or fireman's belay are to guard against #1. These are totally separate issues. Protecting against one doesn't help to protect against the other.
Usurer says:
I am curious if attaching the end of the rope to yourself will be useful and safe way to prevent it from passing through the carabiner. I don't know of any such recommendations but can't think of any bad thing coming from it.
This could work if you coil the rope inside your pack so that it can be neatly paid out. It would have the disadvantage that you would have no chance of doing the thing you normally try to do where you throw the rope down and look to see whether it reached the safe terrain at the bottom of the rappel. I think people sometimes do use this technique in situations where throwing the rope down would be impractical or unworkable, e.g., if it's too windy or the rope would get snagged in brush.
