Have you broken up with someone because of a misunderstanding, an argument, or over something that really shouldn't have caused a break up? Cause if you are then I can understand wanting to fix your relationship even if you had gotten pretty upset with them. There are a few methods you can take first is the active version and after the passive version.
Now to take the active approach and if your ex is willing to talk then this method should be the most straightforward, but if the heat of the last argument hasn't cooled down between the both of you then it most likely will lead to another word fight.
First, come to understand what the argument was about. Then, if the argument is something you regret and that you truly want to get back together then you need to express that with your ex. Now, they might not want to talk to you and still feel upset, but you should kindly tell them that the argument went way out of hand and anything you said wasn't how you really felt and that you would want to possibly talk things over with them. Having sincerity in your words is far more important than pride at this point, but if they won't give you time they certainly don't deserve your attention.
Next we have is the passive method this one focuses solely on self evaluation and to get your ex to look at a fresh you. This one is probably the better option for both you and the entire relationship. The reason I say that is because if you look from how you were before, during, and after the relationship you gotta make some decisions based on how you feel towards yourself. Are you happy with who you are now? If not then you should choose this option, not only is it possible you'll catch your ex's attention, but you will feel better about yourself along with it. The steps for this method is easy. First, try changing your hairstyle to something you normally don't do, but look good with.
Next, spice up your wardrobe with something new for instance try different colored shirts that might bring a different feel that isn't usual for you. Now, the next step should be easier and something you need to think about less before doing and that would be to just hang out with your friends or some new companions and just enjoy yourself.
Now the final step after separating any sad feelings and partly moved on from your ex if you still want to get back together. Talk to your ex with that new found confidence and bright appearance they just might jump at the chance to get back with you. This might make you happy to realize that your hurt feelings from the past are still there and the only reason your standing there now is because of how you feel towards them.
If you can't get past all of the drama with your ex they might not be worth it and then its time to move on and if you take the passive method you will most definitely come out with less scars and a lot more hope for future relationships that you can work on with them and even decide some things up front to prevent past problems to reoccur. I hope my insight is useful for many situations and even help those that need to move on, because there is no fix all answer.