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

Malgorzata Peszynska, Nicholas Slugg

TL;DR

The paper tackles a nonlinear heat equation modeling thawing/freezing in permafrost, where phase change is governed by equilibrium, non-equilibrium, or hysteresis relations between χ and the temperature u. It unifies the analysis by employing a fully implicit time discretization and a double-iteration strategy to handle the two nonlinearities: the phase-function F(u) and the nonlinear diffusion A(U)U, using monotone operator theory and resolvent-based hysteresis components. Theoretical results establish well-posedness and convergence of the discrete schemes for EQ, NEQ, and HYST closures, while numerical experiments on ODEs and PDEs demonstrate robustness, first-order convergence in time, and distinct behaviors across the closures, notably greater effort for HYST. The approach provides a solid computational framework for permafrost modeling, enabling reliable simulations under instantaneous, delayed, or history-dependent phase transitions with potential for parameter identification and scale-appropriate data fusion.

Abstract

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.

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

TL;DR

The paper tackles a nonlinear heat equation modeling thawing/freezing in permafrost, where phase change is governed by equilibrium, non-equilibrium, or hysteresis relations between χ and the temperature u. It unifies the analysis by employing a fully implicit time discretization and a double-iteration strategy to handle the two nonlinearities: the phase-function F(u) and the nonlinear diffusion A(U)U, using monotone operator theory and resolvent-based hysteresis components. Theoretical results establish well-posedness and convergence of the discrete schemes for EQ, NEQ, and HYST closures, while numerical experiments on ODEs and PDEs demonstrate robustness, first-order convergence in time, and distinct behaviors across the closures, notably greater effort for HYST. The approach provides a solid computational framework for permafrost modeling, enabling reliable simulations under instantaneous, delayed, or history-dependent phase transitions with potential for parameter identification and scale-appropriate data fusion.

Abstract

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.
Paper Structure (31 sections, 13 theorems, 97 equations, 5 figures, 3 tables)

This paper contains 31 sections, 13 theorems, 97 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and technical results
  3. Operator equation in a Hilbert space
  4. Evolution equation on $V$ and finite difference approximation
  5. Evolution equation under constraints on $\mathbb{R}$ and generalized play model of hysteresis
  6. Constraint graph on $\mathbb{R}$
  7. ODE with a constraint graph
  8. ODE with a time-dependent constraint in generalized play hysteresis
  9. PDE model with equilibrium, non-equilibrium, and hysteresis variants and its fully discrete approximation
  10. Physical model
  11. Simplified PDE model
  12. Complete simplified model and fully discrete counterpart
  13. Equilibrium system based on \ref{['eq:EQ']}
  14. Non-equilibrium model based on \ref{['eq:NEQ']}
  15. Hysteresis model with \ref{['eq:HYST']}
  16. ...and 16 more sections

Key Result

Theorem 1

Let $T:V \to V$ be (i) strongly monotone and (ii) Lipschitz continuous. Then for any $b\in V$ there is a unique solution $u \in V$ to eq:T. Moreover, if $T(u_1)=b_1,T(u_2)=b_2$ then $||u_1-u_2||_{} \leq \frac{1}{c_0}||b_1-b_2||_{}.$.

Figures (5)

  • Figure 1: Illustration of of relationships \ref{['eq:uchi']}. On left, we plot of $\chi=F(u)$ in \ref{['eq:EQ']}. On right, we plot the same relationship $\chi=F(u)$ again as well as a scatter plot of the results $(u,\chi)$ of simulations+ for \ref{['eq:NEQ']} and \ref{['eq:HYST']}. Details of this example are given in Section \ref{['sec:results-pde']}.
  • Figure 2: The envelopes of the generalized play HYST model calibrated in Example \ref{['ex:hyst-FG']} (i) and (ii). The titles indicate the parameters ($b,\bar{b},\theta_0$).
  • Figure 3: Phase plot of solutions $(U^n,\Upsilon^n)_n$ to Example \ref{['ex:hyst-ODE']} case (ii) and (iii).
  • Figure 4: Solutions to Example \ref{['ex:pde-EQ']} and \ref{['ex:pde-ALL']} at time steps $t=0.5,1,1.5,2,2.5,3$. We plot both the temperature $u(x,t)$ (scale on the left axis) and the water fraction $\chi(x,t)$ (scale on the right axis). In the left column we plot only the solutions to the (EQ) model, as indicated in the legend. In the middle and right we plot the solutions to the (NEQ) and (HYST) models, respectively, with the equilibrium solution (EQ) plotted for reference.
  • Figure 5: Phase plot of the solutions to Example \ref{['ex:pde-ALL']} for $B=5$ (left) and $B=20$ (right).

Theorems & Definitions (29)

  • Theorem 1: AHan2, Thm. 5.1.4, p211
  • Theorem 2: AHan2, Thm. .5.1.3, p209
  • Theorem 3: AHan2, Thm. .5.4.1, p236
  • Theorem 4: Ulbrich, Prop. 2.12, p29, Prop. 2.26, p35
  • Lemma 1
  • proof
  • Lemma 2
  • Remark 1
  • Example 1
  • Remark 2
  • ...and 19 more