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Numerical analysis of the stochastic Stefan problem

Jerome Droniou, Muhammad Awais Khan, Kim Ngan Le

TL;DR

The paper tackles the stochastic Stefan problem with multiplicative noise by employing the gradient discretisation method (GDM) to build a unifying numerical framework. It proves convergence of GDM-based schemes to a weak martingale solution using compactness, Skorokhod representation, and martingale representation arguments, and demonstrates the results with two GDs, MLP1 and HMM. The theoretical analysis is complemented by numerical experiments in two dimensions, showing convergence behavior and revealing that observed mushy regions are largely discretization artifacts that vanish with mesh refinement. Overall, the work provides a rigorous, broadly applicable approach to numerically analyzing stochastic moving-boundary problems and validates it through concrete GD implementations.

Abstract

The gradient discretisation method (GDM) -- a generic framework encompassing many numerical methods -- is studied for a general stochastic Stefan problem with multiplicative noise. The convergence of the numerical solutions is proved by compactness method using discrete functional analysis tools, Skorohod theorem and the martingale representation theorem. The generic convergence results established in the GDM framework are applicable to a range of different numerical methods, including for example mass-lumped finite elements, but also some finite volume methods, mimetic methods, lowest-order virtual element methods, etc. Theoretical results are complemented by numerical tests based on two methods that fit in GDM framework.

Numerical analysis of the stochastic Stefan problem

TL;DR

The paper tackles the stochastic Stefan problem with multiplicative noise by employing the gradient discretisation method (GDM) to build a unifying numerical framework. It proves convergence of GDM-based schemes to a weak martingale solution using compactness, Skorokhod representation, and martingale representation arguments, and demonstrates the results with two GDs, MLP1 and HMM. The theoretical analysis is complemented by numerical experiments in two dimensions, showing convergence behavior and revealing that observed mushy regions are largely discretization artifacts that vanish with mesh refinement. Overall, the work provides a rigorous, broadly applicable approach to numerically analyzing stochastic moving-boundary problems and validates it through concrete GD implementations.

Abstract

The gradient discretisation method (GDM) -- a generic framework encompassing many numerical methods -- is studied for a general stochastic Stefan problem with multiplicative noise. The convergence of the numerical solutions is proved by compactness method using discrete functional analysis tools, Skorohod theorem and the martingale representation theorem. The generic convergence results established in the GDM framework are applicable to a range of different numerical methods, including for example mass-lumped finite elements, but also some finite volume methods, mimetic methods, lowest-order virtual element methods, etc. Theoretical results are complemented by numerical tests based on two methods that fit in GDM framework.
Paper Structure (15 sections, 14 theorems, 137 equations, 29 figures, 2 tables)

This paper contains 15 sections, 14 theorems, 137 equations, 29 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Gradient scheme and main results
  3. Assumptions and notations
  4. Gradient scheme
  5. Examples of gradient discretisations
  6. Main results
  7. A priori estimates
  8. Compactness of the solution to the scheme
  9. Identification of the limit
  10. Numerical Examples
  11. Accuracy tests
  12. Stochastic mushy regions
  13. Conclusion
  14. Acknowledgement
  15. Appendix

Key Result

Lemma 2.6

If a sequence $(\mathcal{D}_{m})_{m\in\mathbb{N}}$ of space-time GDs is compact or limit conforming then it is coercive: there exists a constant $\rho$ such that

Figures (29)

  • Figure 1: $E^m_{\Pi_D \zeta}$
  • Figure 2: $E^m_{\nabla_D \zeta}$
  • Figure 3: $E^m_{\Pi_D \Xi}$
  • Figure 5: Norm of $\Pi_D \zeta$
  • Figure 6: Norm of $\nabla_D \zeta$
  • ...and 24 more figures

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.3: Consistency
  • Definition 2.4: Limit Conformity
  • Definition 2.5: Compactness
  • Lemma 2.6: Coercivity of sequence of GDs
  • Definition 2.7
  • Theorem 2.8
  • Lemma 3.1: Existence and uniqueness for the GS
  • proof
  • Remark 3.2
  • ...and 28 more