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R. Hammer, W. Pötz, A. Arnold:
"A dispersion and norm preserving finite difference scheme with transparent boundary conditions for the Dirac equation in (1+1)D";
Journal of Computational Physics, 256 (2014), 0; S. 728 - 747.

A finite difference scheme is presented for the Dirac equation
in (1+1)D. It can handle space- and time-dependent mass and potential terms and utilizes exact discrete transparent boundary conditions (DTBCs).Based on a space- and time-staggered leap-frog scheme it avo
ids fermion doubling and preserves the dispersion relation of the continuum problem for mass zero (Weyl equation) exactly.
Considering boundary regions, each with a constant mass and
potential term, the associated DTBCs are derived by first applying this finite difference scheme and the n using the Z-transform in the discrete time variable. The resulting constant coefficient difference equation in space can be solved exactly on each of the two semi-infinite exterior domains. Admitting only solutions in
which vanish at infinity is equivalent to imposing outgoing boundary conditions. An inverse Z-transformation leads to exact DTBCs in form of a convolution in discrete time which suppress spurious reflec
tions at the boundaries and enforce stability of the whole space-time scheme.
An exactly preserved functional for the norm of the Dirac spinor on the staggered grid is presented. Simulations of Gaussian wave packets, leaving the computational domain without reflection, demonstrate the quality of the DTBCs numerically, as well as the importance of a faithful representation of the energy-
momentum dispersion relation on a grid

Dirac equation, finite difference, leap-frog, fermion doubling

http://dx.doi.org/10.1016/j.jcp.2013.09.022