The steady-state solution that we seek could be obtained through pure time-stepping using the transient equations described in Application of the Bohm Condition for Ion Fluxes to Surfaces . Such a procedure is very reliable but also very slow. However, because the time-stepping is robust, we employ this procedure to condition the initial iterate in cases when the Newton’s method is not converging. When the modified Newton’s method fails, Twopnt solves the transient equations for a given number of time steps, and then returns to the Newton method using the result of the time-stepping as the new initial iterate. This trial solution will be closer to the steady-state solution, and thus it is more likely to be within Newton’s convergence domain.

The transient equations are given in Homogeneous 0-D Reactor Models . We solve this system of ordinary differential equations using the backward-Euler method. In this method, the time derivatives are approximated by first-order, backwards finite differences. For example, is approximated as

(16–11)

where here the subscript indicates the time step index and represents the size of the time step. All other terms in the governing equation are evaluated at time level . The time-discretized problem is then another system of nonlinear algebraic equations for the dependent variable vector at time level .

To solve the system of equations for each time step we use the same Newton method as we employ in the steady-state problem. However, the transient problem is much more likely to converge; it should always converge for a sufficiently small time step. The objective, then, is to choose a time step that is sufficiently large to make progress toward the steady solution, yet not so large as to introduce convergence difficulties in the transient solution method. Typical time steps range from 1-10 microseconds, although in plasma systems it is often necessary to choose time steps much smaller than this (10-100 nanoseconds).

After solving the specified number of time steps, Twopnt again attempts to solve the steady problem by Newton’s method. If the steady solution fails again the application reverts to time stepping, beginning where it left off at the last time-stepping procedure. Clearly the better the user’s initial estimate of the solution, the less likely the application will have to resort to time stepping. If the application fails to converge in time-stepping, then, the user may either choose smaller time steps, or try a new starting estimate.

The time-stepping solution procedure described here is not particularly sophisticated, since we are neither interested in the accuracy of the transient path nor in solving the transient problem all the way to steady state. We chose to use a method that is relatively inexpensive per step and employs the same Newton algorithm that is used in the steady-state solution. The transient solution therefore serves only to condition the starting estimates for the Newton algorithm, but does not provide accurate solution of the transient process.