I'm studying up on methods for numerically solving the Schrodinger equation. The Schrodinger equation with a zero potential is formally identical to the heat equation in the sense that we just make one of the coefficients in the equation imaginary.
A practitioner of finite-difference methods for the heat equation has to be aware of certain things. From a casual reading of some Wikipedia articles, the main things I've learned are that:
the forward method can be unstable;
Crank-Nicolson has better asymptotics; and
I need to satisfy the Courant condition in the Crank-Nicolson method.
Do any of these facts change if I switch to the Schrodinger equation with a nonzero potential?
In the Schrodinger equation, a kink in the wavefunction contains arbitrarily short wavelengths and therefore has components that propagate with arbitrarily high velocities. Does this mean that the Courant condition can never be satisfied if there is a kink? In a discretized representation, however, there is no clear distinction between a kink and a differentiable point, so is there some adaptive criterion that can be used to set an appropriate time step?