Having studied the fundamentals of quantum mechanics from the perspective of numerical computation it has become obvious that stability is a major drawback in most fast numerical approaches. The best scheme to tackle with the issue is by Visscher whose explicit algorithm is both fast and stable. Accuracy is not my biggest concern since I’m mostly interested in visualization of the quantum mechanical world but the algorithm holds very well. The algorithm is second order accurate, there is some dispersion of the wave packet however. It is also straightforward to extend it to n-dimensional space.
My processing example applet of the 1D time-dependent Schrödinger wave equation with the source code is at http://www.punainen.org/~biotek/visscher1d
Reference to the original paper: P.B. Visscher “A fast explicit algorithm for the time-dependent Schrödinger equation”, Computers in Physics Nov/Dec 1991, 596-8