A NOVEL ITERATIVE SCHEME FOR THE VERY FAST AND ACCURATE SOLUTION OF NON-LTE RADIATIVE-TRANSFER PROBLEMS

Iterative schemes based on Gauss-Seidel (G-S) and optimal successive o verrelaxation (SOR) iteration are shown to provide a dramatic increase in the speed with which non-LTE radiation transfer (RT) problems can be solved. The convergence rates of these new RT methods are identical to those of upper triangular nonlocal approximate operator splitting techniques, but the computing time per iteration and the memory requir ements are similar to those of a local operator splitting method. In a ddition to these properties, both methods are particularly suitable fo r multidimensional geometry, since they neither require the actual con struction of nonlocal approximate operators nor the application of any matrix inversion procedure. Compared with the currently used Jacobi t echnique, which is based on the optimal local approximate operator (se e Olson, Auer, & Buchler 1986), the G-S method presented here is faste r by a factor 2. It gives excellent smoothing of the high-frequency er ror components, which makes it the iterative scheme of choice for mult igrid radiative transfer. This G-S method can also be suitably combine d with standard acceleration techniques to achieve even higher perform ance. Although the convergence rate of the optimal SOR scheme develope d here for solving non-LTE RT problems is much higher than G-S, the co mputing time per iteration is also minimal, i.e., virtually identical to that of a local operator splitting method. While the conventional o ptimal local operator scheme provides the converged solution after a t otal CPU time (measured in arbitrary units) approximately equal to the number n of points per decade of optical depth, the time needed by th is new method based on the optimal SOR iterations is only root 2/2 roo t 2. This method is competitive with those that result from combining the above-mentioned Jacobi and G-S schemes with the best acceleration techniques. Contrary to what happens with the local operator splitting strategy currently in use, these novel methods remain effective even under extreme non-LTE conditions in very fine grids.