There are no "problem solving methods".

Reason being that if there were such a method, we would be able to automatically crank out solutions by simply following it.

The same goes for the "scientific method" (science is also problem solving). For example, Einstein had 2 miracle years (1905 and 1915), but followed "the scientific method" every day of his 40+ year career. If there was a valid scientific method, his output would have been much more smooth over that entire period.

Solutions to problems come from creativity, and improvement in solutions comes from correcting mistakes in the solutions. Both creativity (having ideas) and error-correction can not be prescribed by a method.

Which is a good thing ! It allows novelty, progress, improvement, ...

By the way, there is a method for "problem definition" though: the steps to take to define a problem, i.e. the AS-IS vs the TO-BE, the obstacles in between, the root causes of them, ... But defining a problem is still independent from the idea that will solve it.