Unified analysis of algorithms for equilibrium, non-equilibrium, and hysteresis models of phase transition in permafrost

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.