An implicit ODE-based numerical solver for the simulation of the   Heisenberg-Euler equations in 3+1 dimensions
  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. 
Arnau Pons Domenech, Hartmut Ruhl
physics.comp-ph