  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 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 first applying this finite difference
scheme and then 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 $l_2$ 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 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, as
well as the importance of a faithful representation of the energy-momentum
dispersion relation on a grid.
