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

TL;DR

This paper introduces a highly efficient LU-based algorithm for solving large, sparse linear systems in NLTE plasma modeling, significantly improving speed and accuracy over traditional methods, and discusses criteria for validating configuration averaging.

Contribution

The paper presents a novel LU decomposition algorithm tailored for sparse, large-scale matrices in plasma physics, outperforming existing methods and providing new criteria for configuration average validation.

Findings

The LU-based method is orders of magnitude faster than Gauss elimination.

No convergence or accuracy issues were encountered with the proposed method.

Configuration averaging validity can be assessed using a fictive temperature criterion.

Abstract

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, or X-ray sources. The proper description of these media in stationary cases requires to solve linear systems of thousands or more rate equations. A possible simplification for this arduous numerical task may lie in some type of statistical average, such as configuration or superconfiguration average. However to assess the validity of this procedure and to handle cases where isolated lines play an important role, it may be important to deal with detailed levels systems. This involves matrices with sometimes billions of elements, which are rather sparse but still involve thousands of diagonals. We propose here a numerical algorithm based on the LU decomposition for such linear systems. This method turns out to be orders of magnitude faster than the traditional Gauss elimination. And at variance with alternate methods based on conjugate gradients or minimization, no convergence or accuracy issues have been faced. Some examples are discussed in connection with the krypton and tungsten cases discussed at the last NLTE meeting. Furthermore, to assess the validity of configuration average, several criteria are discussed. While a criterion based on detailed balance is relevant in cases not too far from LTE but insufficient otherwise, an alternate criterion based on the use of a fictive configuration temperature is proposed and successfully tested. It appears that detailed calculations are sometimes necessary, which supports the search for an efficient solver as the one proposed here.