The other day, my computational fluid dynamics instructor was absent and he sent in his PhD candidate to substitute for him. In the lecture he gave, he seemed to indicate several disadvantages associated with various discretization schemes for fluid flow simulations:
Finite Difference Method: It is difficult to satisfy conservation and to apply for irregular geometries
Finite Volume Method: It tends to be biased toward edges and one-dimensional physics.
Finite Element Method: It is difficult to solve hyperbolic equations using FEM.
Discontinuous Galerkin: It is the best (and worst) of all worlds.
Fluctuation Splitting: They are not yet widely applicable.
After the lecture, I tried asking him where he got this information but he did not specify any source. I also tried to get him to clarify what he meant by DG being the "best and worst of all worlds", but couldn't get a clear answer. I can only assume that he came to these conclusions from his own experience.
From my own experience, I can only verify the first claim that FDM is difficult to apply to irregular geometries. For all other claims, I don't have sufficient experience to verify them. I'm curious how accurate these claimed 'disadvantages' are for CFD simulations in general.