Because elves are immortal, when I read this I pictured her literally laying down and going into some kind of long sleep (I thought "green grave" might mean "green as in alive"), and the forest growing around her--and waking up thousands of years later, when "the world is changed."
However, when I poked around the internet, it seems like no one else shared this interpretation, and most people agreed that Arwen was actually dead.
Is her death corroborated anywhere else?
Manwe's ordinance at the end of the First Age was that all those who have even a drop of Mortal blood will be Mortal by nature, but to Earendil and Elwing, and their descendents, he gave a choice: to be counted among Elves or be counted among Men.
This is covered in HoME 5 (The Lost Road) in the section dealing with the end of the 1937 Quenta Silmarillion, and the text was never subsequently rejected or revised by JRR Tolkien, although Christopher Tolkien editorially removed it from the published Silmarillion for unknown reasons.
The existence of Elves: that is of a race of beings closely akin to Men, so closely indeed that they must be regarded as physically (or biologically) simply branches of the same race.
The conclusion is that Half-elves in Tolkien aren't actually a separate species at all. Instead, the offspring of Men and Elves are Men (hence the Princes of Dol Amroth), with the exception of the descendents of Earendil and Elwing, who - once they make their choice - become the type they choose to be.
A slight complication is that choosing to be a Man is irrevocable for both you and your descendents (this is nowhere stated, but otherwise the Heirs of Elros would have had a similar choice, whereas they didn't) but choosing to be an Elf confers the ability to choose on your descendents (otherwise Arwen, Elladan and Elrohir would have already had their choice made for them).
So when Elrond choose immortality at the end of the First Age/start of the Second Age, he actually became an Elf, and likewise Arwen became a Man.
But she wasn't an Elf. She was Half-Elven, and therefore when Elrond departed she had to choose which kindred she belonged to: and for love of Aragorn she chose mortality. So she did die.
Yes and no. She didn't die the way men do. She was an elf and died an elf. What happened was she faded. When elves love a man it is a serious spiritual connection. They are over turning the will of God. Literally their souls become interwined. So when Aragorn died she lived on but the world 'became grey and dull' as Elrond foretold. This means she was missing a part of herself and could not live without Aragorn. Not even her love for her children, who are still alive, can lift her spirit. She moves to Lothlorien and in time her spirt departs from her body. She was in perfect health, like all first born are, but her soul couldn't take the change and woes of the world so it left. Her soul goes to the Halls of Mandros where she can be reborn in Amen. But she will likely stay in the halls till the world is remade to be with her Aragorn.
How old was Lady Arwen, wife of Aragorn in the Lord of the Rings?