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Study of a TPFA scheme for the stochastic Allen-Cahn equation with constraint though numerical experiments

Niklas Sapountzoglou, Aleksandra Zimmermann

TL;DR

This work analyzes a TPFA-based finite-volume discretization of the stochastic Allen-Cahn equation with a set-valued constraint implemented via a Yosida regularization. It introduces a splitting scheme to decouple the nonlinear Yosida term, proving an error bound between the full scheme and the splitting approximation. Through numerical experiments on polygonal domains with Neumann boundaries, it assesses pathwise behavior, expected values, and convergence under varying noise intensities, highlighting that time accuracy is robust for small noise but can degrade for larger noise. The results also show that, while constant-in-space solutions can be preserved under certain conditions, the scheme is not strictly structure-preserving for all realizations, underscoring trade-offs between solvability, regularity, and physical constraints.

Abstract

This contribution provides numerical experiments for a finite volume scheme for an approximation of the stochastic Allen-Cahn equation with homogeneous Neumann boundary conditions. The approximation is done by a Yosida approximation of the subdifferential operator. The problem is set on a polygonal bounded domain in two or three dimensions. The non-linear character of the projection term induces challenges to implement the scheme. To this end, we provide a splitting method for the finite volume scheme. We show that the splitting method is accurate. The computational error estimates induce that the squared $L^2$-error w.r.t. time is of order $1$ as long as the noise term is small enough. For larger noise terms the order of convergence w.r.t. time might become worse.

Study of a TPFA scheme for the stochastic Allen-Cahn equation with constraint though numerical experiments

TL;DR

This work analyzes a TPFA-based finite-volume discretization of the stochastic Allen-Cahn equation with a set-valued constraint implemented via a Yosida regularization. It introduces a splitting scheme to decouple the nonlinear Yosida term, proving an error bound between the full scheme and the splitting approximation. Through numerical experiments on polygonal domains with Neumann boundaries, it assesses pathwise behavior, expected values, and convergence under varying noise intensities, highlighting that time accuracy is robust for small noise but can degrade for larger noise. The results also show that, while constant-in-space solutions can be preserved under certain conditions, the scheme is not strictly structure-preserving for all realizations, underscoring trade-offs between solvability, regularity, and physical constraints.

Abstract

This contribution provides numerical experiments for a finite volume scheme for an approximation of the stochastic Allen-Cahn equation with homogeneous Neumann boundary conditions. The approximation is done by a Yosida approximation of the subdifferential operator. The problem is set on a polygonal bounded domain in two or three dimensions. The non-linear character of the projection term induces challenges to implement the scheme. To this end, we provide a splitting method for the finite volume scheme. We show that the splitting method is accurate. The computational error estimates induce that the squared -error w.r.t. time is of order as long as the noise term is small enough. For larger noise terms the order of convergence w.r.t. time might become worse.

Paper Structure

This paper contains 14 sections, 7 theorems, 54 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The stochastic Allen-Cahn problem with constraint
  3. State of the art
  4. Discretization framework and notation
  5. General notation
  6. Uniform time step and admissible finite-volume meshes
  7. Aim of this study
  8. Splitting method
  9. Numerical experiments
  10. Pathwise properties
  11. General behaviour of the splitting method
  12. Constant solutions
  13. Expected value of the solution
  14. Numerical error estimates

Key Result

Proposition 1.3

BSVZ25 Let $\mathcal{T}$ be an admissible finite-volume mesh of $\Lambda$ in the sense of Definition defmesh with a mesh size $h$. Moreover, let an initial $\mathcal{F}_0$-measurable random vector $u_h^0\equiv (u^0_K)_{K\in\mathcal{T}}\in\mathbb{R}^{d_h}$, $N\in\mathbb{N}$, and $\varepsilon>0$ be gi

Figures (4)

  • Figure 1: Two neighboring control volumes $K,L \in \mathcal{T}$ with interface $\sigma = K|L$ in $\mathbb{R}^2$
  • Figure 2: First graph: squared $L^2$-error with $a=1$. Second graph: squared $L^2$-error with $a=5$.
  • Figure 3: First graph: squared $L^2$-error with $a=30$. Second graph: squared $L^2$-error with $a=60$.
  • Figure 4: squared $L^2$-error with $a=60$ for $N \in N_1 \cup N_2$.

Theorems & Definitions (15)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • ...and 5 more