  With the advent of ultra-high power lasers the nonlinear nature of the vacuum
of quantum electrodynamics (QED) can be probed. Due to the highly nonlinear
structure of the underlying equations new numerical algorithms are required. A
numerical scheme for simulating the nonlinear optical effects of the QED vacuum
in up to 3 spatial dimensions plus time is derived. Its properties are
discussed. The validity of the numerical approach is verified with the help of
known analytic results. The algorithm is used to explore nonlinear all optical
effects of the nonlinear vacuum for which analytic methods are inefficient or
impossible.
