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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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