  The Heisenberg-Euler theory of the quantum vacuum supplements Maxwell's
theory of electromagnetism with nonlinear light-light interactions. These
originate in vacuum fluctuations, a key prediction of quantum theory, and can
be triggered by high-intensity laser pulses, causing a variety of intriguing
phenomena. A highly accurate numerical scheme for solving the nonlinear
equations due to the leading orders of the Heisenberg-Euler weak-field
expansion is presented. The algorithm possesses an almost linear vacuum
dispersion relation even for comparably small wavelengths and incorporates a
nonphysical modes filter. The implemented solver is tested in one spatial
dimension against a set of known analytical results for vacuum birefringence
and harmonic generation. More complex scenarios for harmonic generation are
demonstrated in two and three spatial dimensions.
