In the new 'LotR: A Reader's Companion' Hammond & Scull cite Tolkien's unpublished letter to Eileen Elgar, begun 22 September 1963:
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Tolkien suggests that Arwen could have surrendered her life at the same time as Aragorn, but she was not yet prepared to do so. Although she had become mortal, by nature she was still Elvish, with the long view of life held by that immortal race, to whom 'the gift of the One to Men' ... is bitter to recieve.
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It would seem, therefore, that on becoming mortal she became like Aragorn, with the Numenorean 'grace' to lay down her life when she chose. She just couldn't bring herself to do it at that point.
This puts their parting in a whole new light. It wasn't so much a case of Aragorn 'deserting' Arwen, leaving her alone. It was more a case that he knew it was his time to die while she retained the option to go on - which she took.
So, Arwen died on Cerin Amroth because by that time she was prepared to do so.