Astron. Astrophys. 333, 687-701 (1998)

5. The global iteration procedure

For 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.

Fig. 1. The flow chart of the global iteration procedure

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 y across the whole shock wave model. A typical behaviour of global iterations is shown in Fig. 2 displaying in the upper panel the maximum absolute relative change of the electron temperature and the divergence of radiative flux within the whole shock wave model. On the lower panel of Fig. 2 is shown the iterative change of the total radiative flux emerging ahead the shock wave. Thus, if the initial approximation is good enough, we obtained in all cell centers of the model the exponentially decaying oscillations of the solution vector. The global iterations were stopped when relative changes of variables ceased to decrease. The final relative changes of the solution vector depend on the both spatial resolution of the discrete model and accuracy of integration of differential equations (20) - (24). It should be noted also that the convergence of global iterations appreciably varies for different regions of the shock wave. In particular, the convergence is best in the radiative precursor and thermalization zone, whereas in the recombination zone becomes slower. Thus, the convergence plots shown in Fig. 2 display mostly variations of the solution vector in the vicinity of the outer boundary of the postshock region.

Fig. 2. Upper panel: convergence plots for the electron temperature (solid line) and the divergence of radiative flux (dot-dashed line) given by the maximum absolute relative change . Lower panel: convergence plot for the total radiative flux emerging from the outer boundary of the preshock region

© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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