There are generic methods for solving systems of ODEs numerically. Are there generic methods for PDEs? If so, what are they? If not, why not? To elaborate...
Any set of ODEs can be written in standard form as a set of n first order ODEs in n variables. A single method can then be used. The number of boundary/initial conditions required is n. Not absolutely every ODE with a solution can be solved by a single method and mathematicians will understand advantages of certain methods for certain problems, but science/engineering type of people can expect to use the method to get a solution to a reasonable problem if one exists. Example implementations include bvp4c in MATLAB and scipy.integrate.solve_bvp in Python.
I am not aware of a single version of FEM/FVM or any other method that can be applied as reliably as the ODE methods. But systems of PDEs can also be reduced to a first order standard form and possibly other standard forms. Complications that may prevent a generic method I'm aware of include: behavioural differences based on elliptic/hyperbolic classification (definitely applies to second order PDEs but possibly also any system of PDEs); form of boundary condition required (eg whole-boundary conditions vs Cauchy conditions, which may be directly related to ellipticity); behavioural difference for even/odd derivatives(?) Do any of these things prevent a single generic method? Are there any other reasons?
In MATLAB, solvepde works on systems of second order PDEs, so I think this excludes any PDEs with odd (spatial) derivatives. I'm not sure if the C++/Python FEniCS can represent any system of PDEs with its form language. If it can, my understanding is still that the method will not converge if certain conditions are not met on the choice of basis functions. So it seems these are fairly general methods but not completely generic from the point of view of a science/engineering programmer who just wants a solution without understanding lots of theory (as is the case for the mentioned ODE solvers).