Only a relatively small number of economists use continuous time methods, and an even smaller fraction of them solves these models numerically. It is not taught in most graduate programs.
When the functions are sufficiently smooth, a common choice is collocation, eg with complete polynomials or splines. You need a good initial guess for convergence, linear or higher order perturbations are usually used for this. “Applied Computational Economics and Finance” by Miranda and Fackler has the most detailed treatment of these problems I have seen in economics, could be a good starting point.
Discretizations are the last resort, because they converge slower and it is tricky to get them to be accurate. A very robust set of methods were developed by Kushner and Dupuis (see their textbook by Springer), this basically converts into a discrete time problem with sparse & local transition laws for Ito processes, and you get the usual benefits (contraction, etc).
Value functions can have high curvature in some regions (my understanding is that this is what ODE people call “stiff”, but I may be mistaken). Tricks and transformations can cope with this, and also with non-smooth regions and similar. More of an art than science, as @jlperla said, there are no “standard” methods.