How could I convince my PhD adviser that the method, which is provided by him to me, does not work?
You are in your third PhD year and have done a thorough review of the state of the art. It seems your adviser cannot give you proper reasons for using the existing method and has no relevant papers in this field. This means you are the expert here. If you find the new method superior, then go ahead, implement it and use it. Of course, you have to invest time but eventually it will pay off as you can overcome certain limitations and your simulations will run faster. In the end it is your thesis and if you knowingly stick to a inferior method, you stay below what is possible.

Maybe you can even write a comparison of the two methods in terms of performance (if you are interested). If your adviser accepts the comparison result, then you can use your favourite method and get a first paper out of it.

If your adviser insists on using a method against all scientific proof, then it might be time for a change. Of course, a change of superviser or research group at this advanced stage of your PhD must be considered thoroughly and probably is the last resort.