Highly efficient sparse-matrix inversion techniques and average procedures applied to collisional-radiative codes

The behavior of non-local thermal equilibrium (NLTE) plasmas plays a central role in many fields of modern-day physics, such as laser- produced plasmas, astrophysics, inertial or magnetic confinement fusion devices, and X-ray sources. In steady-state cases the proper description of these plasmas may require the solution of thousands of linear rate equations. A possible simplification for this numerical task lies in some form of statistical averaging, such as the averaging over configurations or superconfigurations. However, to assess the validity of such an averaging procedure and to handle cases where isolated lines play an important role, it will be necessary to treat detailed levels systems. This involves matrices with potentially billions of elements, which are rather sparse but still involve thousands of diagonals above and below the main one. We propose here a numerical algorithm based on the LU decomposition for such linear systems. It will be shown that this method is orders of magnitude faster than the traditional Gauss elimination. Moreover, it is found that there are no convergence or accuracy issues, which are found when using methods based on conjugate gradients or minimization. Among cases treated at the last NLTE-kinetics-code meeting, krypton and tungsten plasmas are considered. Furthermore, to assess the validity of configuration averaging, several criteria are discussed. While a criterion based on detailed balance is relevant in cases not too far from LTE, it is found to be insufficient in general. An alternate criterion based on the inspection of the influence of an arbitrary configuration temperature is proposed and tested successfully.