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In the previous section we derived meshing constraints
necessary for a mass conserving discretization.
We discretize now directly the Wigner equation using
finite differences on an equispaced product-grid in .
Alternative
approaches discretize the Schrödinger or von Neumann equation
and perform some kind of a discrete Wigner transformation. This leads
to difficulties with the domain of the mesh, see [Fre87].
See also [MH94] where the Schrödinger equation is
carefully discretized and a staggered grid is used.
Our implementation follows [Bie97]
and [KKFR89]. Some physical background can be
found in [Fre90].
For the Wigner equation our main interest was in the stationary case
as the Monte Carlo simulator was based on the stationary method.
The discretization of the terms for free streaming, potential
and relaxation time scattering is largely independent from
each other. They are treated separately in the next subsections.

**Subsections**
** Previous:** 8.3.2 Meshing Constraints
**Up:** 8. Finite Difference Wigner
** Next:** 8.4.1 Free Term

R. Kosik: Numerical Challenges on the Road to NanoTCAD