# Eigenvalue Method for NEI Unit in FLASH Code

## TL;DR

This paper introduces an eigenvalue-based NEI method for the FLASH code that enhances computational efficiency, allows easy atomic data updates, and enables simultaneous radiative cooling calculations, improving simulation accuracy and flexibility.

## Contribution

The paper presents a novel eigenvalue approach for NEI calculations in FLASH, offering improved efficiency, easier atomic data updates, and integrated radiative cooling computations.

## Key findings

- Eigenvalue method improves calculation efficiency without accuracy loss.
- Updated atomic data significantly affect NEI results in some cases.
- Simultaneous NEI and radiative cooling calculations are now feasible.

## Abstract

We describe an improved nonequilibrium ionization (NEI) method that we have developed as an optional module for the FLASH magnetohydrodynamic simulation code. The method employs an eigenvalue approach rather than the earlier iterative ordinary differential equation approach to solve the stiff differential equations involved in NEI calculations. The new code also allows the atomic data to be easily updated from the AtomDB database. We compare both the updated atomic data and the methods separately. The new atomic data are shown to make a significant difference in some circumstances, although the general trends remain the same. Additionally, the new method also allows simultaneous calculation of the nonequilibrium radiative cooling, which is not included in the original method. The eigenvalue method improves the calculation efficiency overall with no loss of accuracy. We explore some common ways to present the NEI state with a sample simulation and find that using the average ionic charge difference from the equilibrium tends to be the clearest method.