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```3. Calculate characteristic trajectories and space charge distribution using (2.3)
and (2.5).
4. Solve equation (2.1) with p from step 3.
5. Iterate on steps 3 and 4 until
# and p
are converged.
6. Check that the electric field on the emitter surface meets any applied criteria
e.g. Peek's law.
7. If step 6 is not satisfied, update po and return to step 3.
While solution of the equations in the above manner is relatively straightforward,
there are aspects of the implementation which pose significant difficulties, particularly
if a flexible analysis tool is desired. The characteristic trajectories must be calculated
in time which requires the use of algorithms like adaptive Runge-Kutta [11] in order to
ensure the step size guarantees a certain level of accuracy. The error in the trajectory
increases the longer equations (2.3) are integrated. Given the small size of the emitter
relative to the domain, the step size must be very fine in the vicinity of the emitter to
ensure the trajectory does not bypass the emitter. This problem is made worse by the
high electric field strength near the emitter. In addition to the problems with time
integration, a sufficient number of trajectories must be calculated to ensure sufficient
coverage of the domain.
These problems require an implementation that is very
specific to a given EHD thruster geometry. The routines which calculate trajectories
must be tailored for each thruster and perhaps for a given set of boundary conditions.
This underscores the need for an approach which allows simultaneous solution of the
governing equations.
2.2
Model Problem
Validation of the HDG implementation and assessment of convergence rates requires
an analytical solution.
The simplest geometry to validate the implementation is
concentric cylinders as shown in figure 2-1. While general analytic solutions for this
geometry are complex (see [12]), a simple analytical solution is possible by assuming a
constant electric field. It is clear from the geometry that the solution is axisymmetric
27
```
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