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Heat Conduction with Phase Change in Permafrost Modules of Vegetation Models

David Hötten, Jenny Niebsch, Ronny Ramlau, Walter Zulehner

TL;DR

The paper develops a robust, globally convergent solver for 1D heat conduction with phase change in permafrost by combining a weak enthalpy formulation with finite element discretization and a Katzenelson-based nonlinear solver. It proves that the implicit time-stepping system is a piecewise affine homeomorphism, guaranteeing a unique discrete solution and allowing the Katzenelson method to reach the exact root in a finite number of iterations. The method demonstrates higher accuracy than the industry-standard DECP approach while maintaining comparable computational costs in large-scale LPJmL simulations, and it avoids nonphysical freezing artifacts inherent to some linearized schemes. Overall, the approach is physically grounded, scalable, and well-suited for long-term, heterogeneous soil simulations in DGVMs/LSMs.

Abstract

We consider the problem of heat conduction with phase change, that is essential for permafrost modeling in Land Surface Models and Dynamic Global Vegetation Models. These models require minimal computational effort and an extremely robust solver for large-scale, long-term simulations. The weak enthalpy formulation of the Stefan problem is used as the mathematical model and a finite element method is employed for the discretization. Leveraging the piecewise affine structure of the nonlinear time-stepping equation system, we demonstrate that this system has a unique solution and provide a solver that is guaranteed to find this solution in a finite number of steps from any initial guess. Comparisons with the Neumann analytical solution and tests in the Lund-Potsdam-Jena managed Land vegetation model reveal that the new method does not introduce significantly higher computational costs than the widely used DECP method while providing greater accuracy. In particular, it avoids a known nonphysical artifact in the solution.

Heat Conduction with Phase Change in Permafrost Modules of Vegetation Models

TL;DR

The paper develops a robust, globally convergent solver for 1D heat conduction with phase change in permafrost by combining a weak enthalpy formulation with finite element discretization and a Katzenelson-based nonlinear solver. It proves that the implicit time-stepping system is a piecewise affine homeomorphism, guaranteeing a unique discrete solution and allowing the Katzenelson method to reach the exact root in a finite number of iterations. The method demonstrates higher accuracy than the industry-standard DECP approach while maintaining comparable computational costs in large-scale LPJmL simulations, and it avoids nonphysical freezing artifacts inherent to some linearized schemes. Overall, the approach is physically grounded, scalable, and well-suited for long-term, heterogeneous soil simulations in DGVMs/LSMs.

Abstract

We consider the problem of heat conduction with phase change, that is essential for permafrost modeling in Land Surface Models and Dynamic Global Vegetation Models. These models require minimal computational effort and an extremely robust solver for large-scale, long-term simulations. The weak enthalpy formulation of the Stefan problem is used as the mathematical model and a finite element method is employed for the discretization. Leveraging the piecewise affine structure of the nonlinear time-stepping equation system, we demonstrate that this system has a unique solution and provide a solver that is guaranteed to find this solution in a finite number of steps from any initial guess. Comparisons with the Neumann analytical solution and tests in the Lund-Potsdam-Jena managed Land vegetation model reveal that the new method does not introduce significantly higher computational costs than the widely used DECP method while providing greater accuracy. In particular, it avoids a known nonphysical artifact in the solution.

Paper Structure

This paper contains 19 sections, 8 theorems, 85 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Stefan problem Alexiades1993
  4. Weak Formulation Visintin2008
  5. Discretization
  6. Time discretization
  7. Time Discrete Problem
  8. Space discretization
  9. Fully Discrete Problem
  10. System of nonlinear equations
  11. Solving the nonlinear system
  12. Piecewise affine function theory
  13. Homeomorphism property of the implicit system
  14. Katzenelson algorithm for the implicit system
  15. Results
  16. ...and 4 more sections

Key Result

Lemma 3.2

By specifying for $f,g \in V_h$, we have defined an inner product on $V_h$.

Figures (11)

  • Figure 3.1: Illustration of the auxiliary function $\chi_\epsilon$.
  • Figure 3.2: Illustration of the interpolation operator $\Pi_h$ applied to a function f.
  • Figure 5.1: Illustration of the partitioning of $\mathbb R^2$, by (\ref{['eq:prop_dom_part1']}), (\ref{['eq:prop_dom_part2']}) and (\ref{['eq:prop_dom_part3']}).
  • Figure 5.2: Example iterations of the Katzenelson algorithm. See also Figure \ref{['fig:ill_partition']}.
  • Figure 6.1: Convergence of numerical results to analytical temperature distribution at day 15 with increasing resolution. Temperature is denoted by $u$.
  • ...and 6 more figures

Theorems & Definitions (16)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • remark 1
  • remark 2
  • Proposition 4.1
  • remark 3
  • Definition 5.1
  • Theorem 5.2
  • Theorem 5.3
  • ...and 6 more