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The global iteration procedure

All previous studies treated the radiative transfer as an initial value problem which was solved using the shooting method. As was noted above, the principal obstacle in such an approach is that the solution of the nongrey transfer problem is affected by exponetially growing errors. Such a disadvantage is avoided when the radiation tranfer is treated as a two-point boundary value problem (Feautrier method [7]). The coupling between gas flows and radiation field is taken into account by solution of the two-point radiative transfer boundary value problem and by solution of the initial value problem for the fluid dynamics and rate equations. Integration of the ODE is performed from the outer boundary of the precursor where gas is assumed to be unperturbed and is interrupted at the discontinuous jump, where integration is replaced by the Rankine-Hugoniot relations.


  
Figure: The flow chart of the global iteration procedure [2].
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The global iteration procedure (Fig.[*]) was shown [2] to converge though the convergence interval is too narrow because of the exponential dependence of shock wave properties (e.g. the maximum of the divergence of radiative flux $\nabla\!\!\cdot\!{\bf F}_{\rm r}$)on the shock amplitude. The problem of sensitivity of the global iteration procedure to the initial approximation is resolved when calculations are carried out initially for relativelly weak shocks with M<2 i.e., when LTE assumption can be applied for the initial approximation. The sequence of the shock wave models with gradually increasing amplitude of the jump is obtained when the previous model with somewhat smaller amplitude (in the range 1km/s$\le\Delta v_s\le 5\,$km/s) is used as an initial approximation for the global iteration procedure. On each iteration step, when the solution of the radiation transfer equation is obtained, the integration of ordinary differential equations provides with spatial distibutions of electron temperature Te, gas density $\rho$, the number densities of free electrons ne and atomic level populations ni (see Fig.[*]). If relative changes of all quantities are still larger than the convergence criterion, then iterations are continued. In practice, a few iterations are sufficient. Details of the global iteration procedure are given in [2].

This procedure in general resembles the compute of stellar atmosphere models, nevertheless it contains a number of serious complications. First, atomic level populations are not only in strong departures from LTE but are also in significant departures from statistical equilibrium. Second, unlike the stellar atmospheres, where the divergence of radiative flux is $\nabla\!\!\cdot\!{\bf F}_{\rm r}=0$(the condition of radiative equilibrium), in shock waves the part of the energy of hydrodynamic flow is transformed into radiation and the radiative equilibrium is established only far away from the discontinuous jump. Furthemore, in stellar atmosphere models the total radiative flux is given as one of the boundary conditions, whereas in the shock wave model the emerging flux is obtained from the solution of the problem. Third, the rate equations are stiff and need the special treatment in their solution.

Some preliminary results of this method can be found in [2].


next up previous
Next: Prospectives Up: On the Structure of Previous: The ``asymptotic layering'' method

9/11/1998