If there is a lawsuit or other legal proceeding pending, it will likely settle before trial. A very, very small percentage of cases go to trial these days. When the legal lines have just been drawn, it seems incongruous to stop drawing more legal lines.
If you are going to settle anyway, why not ‘cut to the chase’ and work out a mutually agreeable solution early in the life of the dispute? The longer the dispute drags on, the more money is spent and the more difficult it will be to reach the parties’ original goals. The goals of each side will morph into winning for the sake of winning, instead of the original goal which may have been to resume the parties’ performance under the contract, enforce a warranty, resolve conflicts at work, settle Dad’s estate, and the like.
If the parties don’t want to meet themselves, ask a mediator to be an intermediary. It may be difficult to face your opponent but it is easier than making the matter worse by prolonging it.