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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";
in: "ASC Report 06/2013", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2013, ISBN: 978-3-902627-06-3, 1 - 22.

A nite di erence scheme is presented for the Dirac equation in (1+1)D. It can handle space- and timedependent mass and potential terms and utilizes exact discrete transparent boundary conditions (DTBCs).
Based on a space- and time-staggered leap-frog scheme it avoids 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 rst applying this nite di erence scheme and then using the Z-transform in the discrete time
variable. The resulting constant coe cient di erence equation in space can be solved exactly on each of the two semi-in nite exterior domains. Admitting only solutions in l2 which vanish at in nity 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 reflections 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.

Dirac equation, nite di erence method, leap-frog scheme, dispersion preservation, transparent boundary conditions

http://www.asc.tuwien.ac.at/preprint/2013/asc06x2013.pdf