It really depends on code optimization, model equation and parameter values.
If you make a 2D simulation, it will be enough almost for all purposes and problems with grids less than ~1000x1000. But it will be slow, and the good simulation speed is possible only with implicit schemes permits large time-steps.
As for 3D-simulation, it is almost impossible at reasonable grid size even You use a top newest CPU with OpenMP or other multicore parallelism. Only CUDA/OpenACC can give a good results with desktop workstation now, otherwise cluster is better.
It depends on the computational domain. Smaller the mesh size, more is the processing power required. i7 is recommended. Otherwise, u will end up loading the mesh forever.