The need to use an equispaced grid in is a performance
killer. For resonant tunneling diodes one gets sharp resonances
which have to be resolved by a fine enough
-mesh. Although
the equation is sparse in
, the terms for potential
and for scattering give full
blocks
in the matrix.
For self-consistent simulations one usually chooses
where
is given by 8.15.
Then
the completeness condition shows the following
unpleasant feature: if one refines
the mesh everywhere in
and also doubles the number of points
,
then the resolution
in
is unchanged. The
-mesh is only extended towards infinity,
but not denser.
The new
-points are introduced in external areas where the density
is very low and accuracy in
is not increased.
Hence in a large part of the simulation domain the density
is very low as depicted in Figure 8.1.
In this case one has to increase the number of points in
by
a factor of
to achieve double resolution.
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If the nodes are ordered properly, the
overall sparsity structure of the
matrix is a banded block matrix with bandwidth of an order like
where
depends on the chosen discretization
for the free term (that is, a small number, for example
).
This Wigner system can be conveniently solved with a (sparse)
Gaussian elimination solver.
Note that the bandwidth for the Wigner system is approximately
of the same order as for the matrix
stemming from a two dimensional diffusion
problem (Laplace equation).
So with respect to the use of sparse Gaussian solvers, the Wigner
equation is no worse than a ``completely" sparse problem.
But a two-dimensional diffusion problem can conveniently be solved
by iterative solvers, while the low overall sparsity
(full blocks) of the Wigner system makes this slow.
Numerical costs for Gaussian elimination applied
on a
matrix block are of the order of
. The number of such
blocks in the system matrix is proportional to
. Hence
the costs are of the order of
which
is approximately
. Even in one spatial
dimension this sets a tight practical limit on
the feasible size of the mesh. The Wigner function method
is compared with the QTBM in Section 10.1 .
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