In my experience, with CFD, in general we instead choose Finite Volume Methods, as that is a conservative method and guarantees conservation of fields. Of course that is the "traditional" Mechanical Engineering textbook reply. Lately other methods like Lattice-Boltzmann and Immersed Boundary...
I was pursuing a PhD in Mechanical Engineering, but I had to unfortunately drop out due to health reasons. Now I'm working in the financial sector. I have a broad interest in Physics, but also in things that relate to Engineering, because there is frankly a lot of hand waving in Engineering...