  The technique of continuous unitary transformations has recently been used to
provide physical insight into a diverse array of quantum mechanical systems.
However, the question of how to best numerically implement the flow equations
has received little attention. The most immediately apparent approach, using
standard Runge-Kutta numerical integration algorithms, suffers from both severe
inefficiency due to stiffness and the loss of unitarity. After reviewing the
formalism of continuous unitary transformations and Wegner's original choice
for the infinitesimal generator of the flow, we present a number of approaches
to resolving these issues including a choice of generator which induces what we
call the "uniform tangent decay flow" and three numerical integrators
specifically designed to perform continuous unitary transformations efficiently
while preserving the unitarity of flow. We conclude by applying one of the flow
algorithms to a simple calculation that visually demonstrates the many-body
localization transition.
