Is there a general guide to Julia's PDE solving utilities based on the properties of the PDE in question?

Specific Domains Numerics
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As I continue my foray into Julia, I’d like to apply it more directly to my research by utilizing its PDE-solving capabilities to solve the radial Schrodinger equation for one and two-electron systems, with an eye towards eventually replacing more of my inherited Fortran code, which, while functional, is also horribly documented and a pain to maintain and extend. If Julia can provide comparable performance while being much easier to use and modify, that would be amazing. The only difficulty is that there’s a lot of options for PDE solvers in Julia, and it’s not clear from their overviews which ones will be appropriate or performant for my problems. I would like to figure that out before committing to learning a particular solver package and its interface. For reference (though I emphasize that I’m more interested in guidance than a straight “just use this” answer), the radial Schrodinger equation for hydrogen is:

(-\frac{1}{2} \frac{d^2}{dr^2} + \frac{l(l+1)}{2r^2} - \frac{1}{r})u_{nl}(r) = E_n u_{nl}(r)

That’s an ODE, of course (and specifically an eigenvalue problem), but it quickly becomes a PDE if you add a second particle (so you have r_1 and r_2) and/or a time-dependent potential (which adds V(r, t) to the left-hand side and replaces the E_n on the right-hand side with i\frac{\partial}{\partial t}. (Fussy details for those who care: this is in atomic units, I’m ignoring the reduced mass, and the method I would try for say, helium is expanding the angular solution in bipolar spherical harmonics and solving the radial equation for a number of the resulting partial waves). Anyways, if there’s a guide (either specifically for Julia or just in general) for matching the characteristics of a given PDE problem to a particular kind of solution method, please let me know.

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I would take Julia out of the picture for a moment — do you understand the different categories of methods for solving PDE problems and their pros and cons? Spectral methods, finite difference, finite elements, …?

Regardless of programming language, PDE solvers can’t really be treated as a black box — you need to learn something about how they work in order to choose and use them effectively.

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I have a vague sense of different methods—finite difference is straightforward but low-resolution, spectral methods like grid-Fourier and grid-Legendre are global and thus dense grids are computationally expensive, and finite elements introduce more complexity to the setup, but let you use local basis functions with variable grids—but I don’t have a good read of “if PDE has x features, then choose y method.”