Analysis, thermodynamics, and a numerical solver for a pressure-temperature equilibrium closure of the four-equation model
Abstract
We analyze an often used closure model for multi-material hydrodynamics where pressure-temperature equilibrium (PTE) is assumed for every state; emphasis is placed on tabular equations of state. This multi-material model is often referred to as the four-equation model. The identification of the admissible set is presented and is proven to be convex, setting the foundation for development of invariant-domain preserving methods for this model. A novel numerical method is presented for solving the highly nonlinear system for the equilibrated pressure and temperature with an arbitrary number of materials. This new method is compared with some traditional iterative solvers through a collection of different tests. Additionally, we provide a detailed analysis of the thermodynamics of the mixture model for general equations of state and prove existence and uniqueness of the pressure-temperature equilibrium solution under some thermodynamic assumptions.
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BibTeX
@misc{clayton2026analysis,
title = {{Analysis, thermodynamics, and a numerical solver for a pressure-temperature equilibrium closure of the four-equation model}},
author = {Bennett Clayton and Joshua McConnell and Clell Solomon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.27726},
eprint = {2606.27726},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.27726},
}
RIS
TY - UNPB
TI - Analysis, thermodynamics, and a numerical solver for a pressure-temperature equilibrium closure of the four-equation model
AU - Bennett Clayton
AU - Joshua McConnell
AU - Clell Solomon
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2606.27726
UR - https://arxiv.org/abs/2606.27726
AB - We analyze an often used closure model for multi-material hydrodynamics where pressure-temperature equilibrium (PTE) is assumed for every state; emphasis is placed on tabular equations of state. This multi-material model is often referred to as the four-equation model. The identification of the admissible set is presented and is proven to be convex, setting the foundation for development of invariant-domain preserving methods for this model. A novel numerical method is presented for solving the highly nonlinear system for the equilibrated pressure and temperature with an arbitrary number of materials. This new method is compared with some traditional iterative solvers through a collection of different tests. Additionally, we provide a detailed analysis of the thermodynamics of the mixture model for general equations of state and prove existence and uniqueness of the pressure-temperature equilibrium solution under some thermodynamic assumptions.
ER -