Analytical and numerical solutions to the non-diffusive Stefan problem
Abstract
In this work, the Maxwell--Cattaneo--Vernotte (MCV) equation is used to model the one-dimensional hyperbolic Stefan problem in the limit of a small Stefan number (Ste ≪ 1). The solutions are approximated with perturbation series expansions using a reformulation in which time is expressed as a function of the solid-liquid interface position. The first proposed solution is derived in a framework that considers diffusive heat transfer at the phase change interface, for analytic tractability. Two rectification strategies are proposed to address the asymptotic divergence present in this formulation: a rescaled inner solution which is then combined with the outer solution to yield a composite solution, and size-dependent thermo-physical system parameters for better capture of hyperbolic effects at the phase change interface. The resulting interface profiles exhibit a characteristic parabolic-like shape, consistent with diffusive Stefan problem findings, with pronounced early-time hyperbolic effects at larger thermal relaxation times. Parametric studies are done over three pertinent variables in the dimensionless system: the Stefan number (Ste), the dimensionless thermal relaxation time (τ̃), and the thermal diffusivity (α). The studies suggest that model error scales with the Stefan number in accordance with the theoretical truncation error of the perturbation expansion. Additionally, larger values of τ̃ amplify early-time hyperbolic effects, thereby increasing model error, while larger α extends the relative temporal domain over which these hyperbolic effects remain significant, also corresponding to an increase in model error.
Related
- A complete classification of singularities for radial solutions of the Stefan problem
- Numerical Study of Reported Multiple Solutions to a Cauchy Problem for the 2d Incompressible Euler Equations
- Existence of weak solutions to two-phase Stefan problems for parabolic partial differential equation system
- The generic structure of the singular set in the Stefan problem
- Global Strong Solutions for the Ternary Maxwell-Stefan Diffusion System
BibTeX
@misc{ham2026analytical,
title = {{Analytical and numerical solutions to the non-diffusive Stefan problem}},
author = {Matthew Van Ham and Minghan Xu and Samuel Huberman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.12102},
eprint = {2607.12102},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.12102},
}
RIS
TY - UNPB
TI - Analytical and numerical solutions to the non-diffusive Stefan problem
AU - Matthew Van Ham
AU - Minghan Xu
AU - Samuel Huberman
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2607.12102
UR - https://arxiv.org/abs/2607.12102
AB - In this work, the Maxwell--Cattaneo--Vernotte (MCV) equation is used to model the one-dimensional hyperbolic Stefan problem in the limit of a small Stefan number (Ste ≪ 1). The solutions are approximated with perturbation series expansions using a reformulation in which time is expressed as a function of the solid-liquid interface position. The first proposed solution is derived in a framework that considers diffusive heat transfer at the phase change interface, for analytic tractability. Two rectification strategies are proposed to address the asymptotic divergence present in this formulation: a rescaled inner solution which is then combined with the outer solution to yield a composite solution, and size-dependent thermo-physical system parameters for better capture of hyperbolic effects at the phase change interface. The resulting interface profiles exhibit a characteristic parabolic-like shape, consistent with diffusive Stefan problem findings, with pronounced early-time hyperbolic effects at larger thermal relaxation times. Parametric studies are done over three pertinent variables in the dimensionless system: the Stefan number (Ste), the dimensionless thermal relaxation time (τ̃), and the thermal diffusivity (α). The studies suggest that model error scales with the Stefan number in accordance with the theoretical truncation error of the perturbation expansion. Additionally, larger values of τ̃ amplify early-time hyperbolic effects, thereby increasing model error, while larger α extends the relative temporal domain over which these hyperbolic effects remain significant, also corresponding to an increase in model error.
ER -