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Q. What does Donnie do to save the Universe, in detail?

This depends on your interpretation of the film as it has many levels.

In my favoured interpretation the evidence e.g. the Philosophy of Time Travel (which can also be found within the puzzles at DonnieDarko.com, and on some of the DVD versions), points to the fact that the Universe has become corrupted, forming a Tangent Universe. As mentioned above, the logical conclusion is that the event that causes this is the arrival of the jet engine.

Therefore Donnie's objective, through manipulation by Frank (the man in the rabbit costume who appears mostly in his dreams), is to give a valid reason for the engine's appearance i.e. either opening or creating a wormhole back to the past. It is also assumed that he does this with telekinesis and his power to create water (the fourth dimensional construct) as it is stated these are some of the powers of the Living Receiver. So in short, corruption = bad, wormhole = good, hence Universe saved.

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Relevant quotes from the Philosophy of Time Travel that support this:

"Artifacts returned to the Primary Universe are often linked to religious iconography; as their appearance on Earth seems to defy logical explanation."

"The Fourth Dimension of Time is a stable construct, though it is not impenetrable. ....Incidents when the fabric of the fourth dimension becomes corrupted are incredibly rare. ....If a Tangent Universe occurs, it will be highly unstable, sustaining itself for no longer than several weeks."

"....The Manipulated Dead will often set an Ensurance Trap for the Living Receiver to ensure that the Artifact is returned safely to the Primary Universe."

"If the Ensurance Trap is successful, the Living Receiver is left with no choice but to use his Fourth Dimensional Power to send the Artifact back in time into the Primary Universe before the Black Hole collapses upon itself."

"The Living Receiver is often (blessed) with a Fourth Dimensional Power(s). These include increased strength, telekinesis, mind control, and the ability to conjure fire and water."

"The Fourth Dimensional Construct is made of Water." * http://ruinedeye.com/cd/time2.htm