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In The Deathly Hallows (both the book as well as the movie), Ron gets corrupted by the power of the Locket's curse. He has a breakdown with Harry and decides to storm out and leave.
At this point however, Hermione, the love of his life, chooses to stay with Harry for the greater good of the world and to finish the mission they had started.
What compelled Hermione to stay with Harry and not leave with Ron?
Hermione is a woman who, for the most part, does things deliberately, having thought out what, why and how. That is not to say that she doesn't have emotions, only that she doesn't allow them to drive what she does.
For example, before leaving with Harry and Ron at the beginning of The Deathly Hallows, she goes to rather extreme lengths to prepare in every way that she can. She erases herself from her parents' life, sending them to Australia where they will be safe. She also carefully packs a magically expanded purse with a wide variety of "essentials" including a small library.
When Ron leaves Harry and Hermione, he is ruled by his anger and frustration, and his departure is an act of passion. He later tells Harry that he almost instantly regretted leaving and worked for weeks to find a way to rejoin them.
Hermione understands why Ron acted as he did, but she would never have left with him for two reasons. First, as already mentioned, she rarely if ever, loses control of her actions like Ron did. Second, she understands, unlike Angry Ron, that no matter how much frustration or even anger she might be feeling, Harry is still the best hope for defeating Voldemort. Hermione knows that if Voldemort is not defeated that she and Ron have no real hope of any kind of life together. Her love for Ron, her parents, and her other friends would never allow her to put their mission in jeopardy while any chance of success existed.
In my opinion, you've already answered your own question:
for the greater good of the world and to finish the mission they had started.
This isn't the first time Hermione has chosen to do what she believes to be the Right Thing despite how others might view her for doing so (e.g. the Firebolt).
This also isn't the first time Ron has broken from the others over something stupid, only to eventually realize his mistake and make up with them (e.g. Harry being selected for the Triwizard championship); he even has a good excuse this time!
I think it would feel rather out of character for Hermione to abandon the quest to go chase after a fuming Ron.
There are two reasons.
Harry needed her more than Ron because finding the Horcrux was important. Harry alone would not be able to find and destroy them, as she told Harry in the 6th book.
She knew that Ron is doing this because of the influence of the locket, and he will come back after he understands this. Ron was making this decision in anger.
She loved Ron, but Harry and the friendship between them was also important. She values friendship and loyalty, as we all know.
Hermione's decisions are completely logical and positive in the series.
Hermione was not one for hasty decisions. The bad mood had been amplified by the horcrux which she must have been aware off. When Ron has his breakdown she is given the choice to either follow her crush who is leaving in magically amplified rage or stay with her best friend and continue the mission she since long had excepted. She had always been able to see truth and reason when other people didn't, for instance triwizard tournament. To go with Ron would be a very non Hermione thing to do.
