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We normalise the unknown moments in terms of the measured ones as:
 |
(4.1) |
 |
(4.2) |
 |
(4.3) |
Where
is the thermal speed, from which the temperature can be inferred:
 |
(4.4) |
The inputs to the solver are the normalised quantities
 |
(4.5) |
 |
(4.6) |
 |
(4.7) |
For the inputs
,
,
, the exact solution yields unity for equations 4.5 & 4.6 and
for equation 4.7.
The set of non-linear equations which must be inverted are:
 |
(4.8) |
 |
(4.9) |
 |
(4.10) |
That is, the measured moments are functions of the real moments. We recast to a normalised system such that we wish to find the triplet (
,
,
), given the normalised inputs (
,
,
). The equations to be solved are
 |
(4.11) |
![$\displaystyle \zeta_{sc}^2-\eta\int_{V_L}^{V_U} (V^2-\epsilon_{sc}) \left[E^- + E^+
-\frac{V_{Te}^{\prime 2}(E^-E^+)}{2VV_e^\prime\zeta_{sc}}\right]{\rm d}V = 0$](img66.png) |
|
|
(4.12) |
 |
|
|
(4.13) |
where
![\begin{displaymath}
E^{\pm}=\exp{\left[-\left(\frac{V\pm V_e^\prime\zeta_{sc}}{V_{Te}^\prime}\right)^2\right]}
\end{displaymath}](img68.png) |
(4.14) |
Next: Adding high energy moments
Up: The algorithm
Previous: Choice of frame
Contents
Steve Schwartz
2005-03-26