Numerical treatment of the energy equation in compressible flows simulations

TL;DR

This paper investigates how different discretizations of the energy equation in compressible Euler equations affect conservation properties, analyzing both theoretical and numerical aspects to identify effective schemes for shock-free flows.

Contribution

The study provides a comprehensive analysis of various energy discretization methods, including new formulations, highlighting their conservation properties and performance in test cases.

Findings

Locally conservative, kinetic-energy preserving schemes improve accuracy.

Different energy formulations influence conservation and stability.

New formulations show potential advantages in test simulations.

Abstract

We analyze the conservation properties of various discretizations of the system of compressible Euler equations for shock-free flows, with special focus on the treatment of the energy equation and on the induced discrete equations for other thermodynamic quantities. The analysis is conducted both theoretically and numerically and considers two important factors characterizing the various formulations, namely the choice of the energy equation and the splitting used in the discretization of the convective terms. The energy equations analyzed are total and internal energy, total enthalpy, pressure, speed of sound and entropy. In all the cases examined the discretization of the convective terms is made with locally conservative and kinetic-energy preserving schemes. Some important relations between the various formulations are highlighted and the performances of the various schemes are assessed by considering two widely used test cases. Together with some popular formulations from the literature, also new and potentially useful ones are analyzed.