In this paper we consider a nonlinear partial differential equation describing heat flow with ice-water phase transition in permafrost soils. Such models and their numerical approximations have been well explored in the applications literature. In this paper we describe a new direction in which the allow relaxation and hysteresis of the phase transition which introduce additional nonlinear terms and complications for the analysis. We present numerical algorithms as well as analysis of the well-posedness and convergence of the fully implicit iterative schemes. The analysis we propose handles the equilibrium, non-equilibrium, and hysteresis cases in a unified way. We also illustrate with numerical examples for a model ODE and PDE.

翻译：本文研究描述多年冻土中冰水相变热传导过程的非线性偏微分方程。此类模型及其数值近似方法在应用文献中已有充分探讨。本文提出新的研究方向：允许相变的弛豫与滞后效应，这引入了额外的非线性项并增加了分析复杂性。我们提出了数值算法，并对全隐式迭代格式的适定性与收敛性进行了分析。所提出的分析方法以统一方式处理平衡、非平衡及滞后三种情形。同时通过模型常微分方程与偏微分方程的数值算例进行了演示。