Table I summarizes the results of this analysis of the discrete open-system model. It also contains results for other discretization schemes which have been used for similar calculations.
The schemes included in the Table are, first, the present
upwind-difference approximation to the 
operator, denoted
``Upwind.'' Second is the centered-difference approximation studied by
Jensen and Buot (1989a) to resolve the problem of detailed
balance. In this approximation the kinetic-energy superoperator
(for centered-difference) becomes
The third column presents an analysis of a centered-difference
approximation with upwind differencing applied only at the outflowing
boundaries. This is the 
limit of the Lax-Wendroff
discretization (with upwind differencing at the boundaries) used by
Kluksdahl, Kriman, Ferry, and Ringhofer (1989).
The continuous time limit is
invoked in the present analysis to remove any artifacts of the time
discretization and thus evaluate this scheme on the same basis as the
others. This yields the superoperator 
(for
centered, upwind boundary):
The last column of Table I summarizes the time-reversible model based upon the density matrix that we explored in Section 3.
The errors in the continuity and momentum balance relations were
determined by analysis of the discrete equations in the manner described
above. These errors include contributions from the potential
superoperator 
as well as from the kinetic-energy superoperator
. Because the different discretization schemes that can be used for
are independent of those for 
, the error
contributions from 
(denoted as 
) discussed in Section
2 are tabulated separately in Table II.
The density matrix model of Section 3 is set up so as to
exactly satisfy the continuity and momentum balance equations. This is
possible because the 
superoperator can be evaluated in
closed form when applied to the density matrix in real space, but must
be approximated by (4.43), (7.122), or some similar
expression when applied to a Wigner function.
The centered difference form (7.125) also exactly satisfies
the continuity and momentum balance equations, if we associate the
current density J with the mesh points rather than with the intervals
as in (7.114). In the case of the centered, upwind boundary
scheme (7.126) the change in the discretization of the gradient
necessarily introduces errors of 
into all the moment
equations. It can be argued that such errors are in some way less
significant because they only occur adjacent to the boundaries, but a
central lesson of the present analysis is that the boundary terms affect
the entire solution, and their influence is not localized to the regions
near the boundaries.
The considerations which bear upon the departures from detailed
balance have been discussed above. The approach described, studying the
scaling properties of the equilibrium solutions to the Liouville
equation as illustrated in Fig. 23, does not work for the
centered-difference (7.125)
or centered-upwind boundary (7.126) discretizations because
one cannot directly solve for the
steady-state distributions with these schemes. Both of them possess at
least one
spurious mode whose eigenvalue is very close to zero, which in regions
of constant potential is of the form 
, so that its sign
alternates between adjacent mesh points. If one attempts to solve for
the steady-state distribution, a relatively arbitrary fraction of this
mode is incorporated into the solution, rendering the results
meaningless. Nevertheless, the considerations previously discussed
strongly suggest that the discretization (7.126), at least,
probably violates detailed balance to the same order as the
upwind-difference scheme. The status of the centered-difference scheme
(7.125) is more problematical. Jensen and Buot (1989a) obtained
improved results in the sense of a small equilibrium current with this
scheme, but it does not seem to be particularly
distinguishable from the others on the basis of the symmetry property
(7.124) or its commutator with 
.
The density matrix approach is
presumed to satisfy detailed balance exactly because it is
time-reversible.
The stability condition is, of course, absolutely
essential for a useful model. It is expressed in Table I by the scaling
order of the greatest imaginary part of an eigenvalue.
The scaling properties of the different discretizations were
investigated by a procedure similar to that illustrated in
Fig. 23. Both the upwind-difference and centered-upwind
boundary schemes are stable, as we expect (all imaginary parts are
negative), but the scaling is different. This is illustrated in
Fig. 24, which shows the eigenvalue spectrum for the
centered-upwind boundary scheme for the same structure used previously.
While all the eigenvalues lie in the lower half-plane, they are
clustered much nearer the real axis than those of the upwind scheme
illustrated in Fig. 9. The centered-difference and the
density matrix schemes are not stable, as they possess eigenvalues with
positive imaginary parts. (It should be noted that the specific results
obtained for the
centered-difference scheme are somewhat suspect. The 
dependence is suspiciously close to that of the total number of
arithmetic operations required to diagonalize the operator, 
,
so there is a strong possibility that what was observed here is just the
cumulative effect of roundoff errors.)
Figure 24. Eigenvalue spectrum resulting from the discretization (7.126). This discretization results in a stable model.
In summary, no model exactly satisfies all the conditions one would desire. One must therefore decide which model to use on the basis of what is most important for a given application. The information in Tables I and II provides the basis for making such a decision. The analyses which are summarized in the Tables, while somewhat tedious, will be useful at two different levels. The first is as a summary of the properties of the different discretization schemes studied here. At a more general level the present analyses provide an example of the sort of study which is required to make sense of the multitude of discretization schemes for a given physical problem.
