Linear and entropy-stable scheme for the dynamic van der Waals model of non-isothermal liquid-vapor flows
Abstract
In this paper, we propose a linear and decoupled fully discrete scheme that preserves the entropy production property for the dynamic van der Waals model. This model provides a thermodynamically consistent diffuse-interface framework for describing non-isothermal phase transitions in single-component liquid-vapor systems, containing the coupled conservation laws of mass, momentum and energy. We first introduce an equivalent reformulation of the governing equations that avoids the use of chain-rule identities, which generally fail under discrete differentiation. Further, a component-wise convex-splitting method is employed to calculate the discrete chemical potential and temperature. This treatment provides a discrete entropy-stable inequality without a time step restriction. Based on the explicitly treated velocity, the scheme allows the three conservation equations to be solved linearly and sequentially in a decoupled manner. Since the momentum equation does not directly affect the entropy-stability analysis, we use a classical linear semi-implicit scheme to solve it. Numerical experiments verify the expected numerical accuracy. Simulations of droplet evaporation and two- and three-dimensional Leidenfrost effects further demonstrate the proposed method can capture complex non-isothermal phase-transition dynamics while numerically reproducing the discrete entropy production behavior.
Related
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BibTeX
@misc{guo2026linear,
title = {{Linear and entropy-stable scheme for the dynamic van der Waals model of non-isothermal liquid-vapor flows}},
author = {Yunzhuo Guo and Zhonghua Qiao and Xianmin Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13029},
eprint = {2610.13029},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13029},
}
RIS
TY - UNPB
TI - Linear and entropy-stable scheme for the dynamic van der Waals model of non-isothermal liquid-vapor flows
AU - Yunzhuo Guo
AU - Zhonghua Qiao
AU - Xianmin Xu
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13029
UR - https://arxiv.org/abs/2610.13029
AB - In this paper, we propose a linear and decoupled fully discrete scheme that preserves the entropy production property for the dynamic van der Waals model. This model provides a thermodynamically consistent diffuse-interface framework for describing non-isothermal phase transitions in single-component liquid-vapor systems, containing the coupled conservation laws of mass, momentum and energy. We first introduce an equivalent reformulation of the governing equations that avoids the use of chain-rule identities, which generally fail under discrete differentiation. Further, a component-wise convex-splitting method is employed to calculate the discrete chemical potential and temperature. This treatment provides a discrete entropy-stable inequality without a time step restriction. Based on the explicitly treated velocity, the scheme allows the three conservation equations to be solved linearly and sequentially in a decoupled manner. Since the momentum equation does not directly affect the entropy-stability analysis, we use a classical linear semi-implicit scheme to solve it. Numerical experiments verify the expected numerical accuracy. Simulations of droplet evaporation and two- and three-dimensional Leidenfrost effects further demonstrate the proposed method can capture complex non-isothermal phase-transition dynamics while numerically reproducing the discrete entropy production behavior.
ER -