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.