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

An implicit ODE-based numerical solver for the simulation of the Heisenberg-Euler equations in 3+1 dimensions