{'fraction_non_alphanumeric': 0.01748971193415638, 'fraction_numerical': 0.0, 'mean_word_length': 5.852112676056338, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Solving two‐phase freezing Stefan problems: Stability and monotonicity

The two‐phase Stefan problems with phase formation and depletion are special cases of moving boundary problems with interest in science and industry. In this work, we study a solidification problem, introducing a front‐fixing transformation. The resulting non‐linear partial differential system involves singularities, both at the beginning of the freezing process and when the depletion is complete, that are treated with special attention in the numerical modelling. The problem is decomposed in three stages, in which implicit and explicit finite difference schemes are used. Numerical analysis reveals qualitative properties of the numerical solution spatial monotonicity of both solid and liquid temperatures and the evolution of the solidification front. Numerical experiments illustrate the behaviour of the temperatures profiles with time, as well as the dynamics of the solidification front.