Extracting the Temperature Analytically In Hydrodynamics Simulations with Gas and Radiation Pressure

TL;DR

This paper presents an analytical solution for extracting temperature in hydrodynamics simulations involving gas and radiation pressure, replacing traditional iterative methods with explicit formulas for improved efficiency and accuracy.

Contribution

It introduces an analytical approach to solve the quartic equation for temperature, offering explicit expressions and comparison with iterative methods in hydrodynamics simulations.

Findings

Analytical solutions for temperature extraction are derived.

Explicit formulas improve computational efficiency.

Comparison shows advantages over iterative methods.

Abstract

Numerical hydrodynamics simulations of gases dominated by ideal, nondegenerate matter pressure and thermal radiation pressure in equilibrium entail finding the temperature as part of the evolution. Since the temperature is not typically a variable that is evolved independently, it must be extracted from the the evolved variables (e.g. the rest-mass density and specific internal energy). This extraction requires solving a quartic equation, which, in many applications, is done numerically using an iterative root-finding method. Here we show instead how the equation can be solved analytically and provide explicit expressions for the solution. We also derive Taylor expansions in limiting regimes and discuss the respective advantages and disadvantages of the iterative versus analytic approaches to solving the quartic.