## 5. The global iteration procedureFor integration of the system of ordinary differential equations (20) - (24) we have to know the mean intensity and the divergence of radiation flux at each cell center of the shock wave model. On the other hand, the radiation transfer equation (50) can be solved only when the extinction and emission coefficients are given in each cell center. In order to take into account a coupling between gas material and radiation field we employ the iteration procedure. If the initial approximation is enough close to the final solution, we may hope that each iteration will give a better approximation for the final solution than the previous one. In Fig. 1 is shown the flow chart where the main steps of the global iteration procedure are depicted.
Within the whole shock wave both the radiation field and atomic level populations are in a strong departure from LTE. This feature is the principal difficulty accompanying the shock wave model calculations because it is responsible for the narrow convergence area of global iterations. In particular, the use of the initial LTE approximation allows the converged solution to be obtained only for the weak shock waves with upstream Mach numbers . For larger upstream velocities the initial oscillation amplitude of the solution vector becomes so large that some quantities fall beyond their physical meaning. In order to alleviate this difficulty and to be able to consider the structure of stronger shock waves we computed a grid of the shock wave models with gradually increasing upstream Mach number, the LTE initial approximation being used only for the first model with the Mach number . Thus, each model of the grid with exception of the first one was computed with initial approximation obtained from the previous converged model having somewhat smaller upstream velocity. The upstream velocity increment was in the range . For test purposes some models were computed with different initial conditions taken from shock wave models with both larger and smaller upstream velocities. In all such cases the shock wave structure was found to converge to s single solution. In order to describe the convergence of global iterations we introduce for each component of the solution vector at the -th iteration the quantity specifying the maximum absolute relative change of the variable
© European Southern Observatory (ESO) 1998 Online publication: April 20, 1998 |