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Metastable Speeds in the Fractional Allen-Cahn Equation

Franz Achleitner, Christian Kuehn, Jens Markus Melenk, Alexander Rieder

TL;DR

It is investigated how the width and speed of the interfaces change if the authors vary the exponent $\alpha$ of the fractional Laplacian, and asymptotic formulas for the interface speed and time-to-collision give a good approximation for large intervals.

Abstract

We study numerically the one-dimensional Allen-Cahn equation with the spectral fractional Laplacian $(-Δ)^{α/2}$ on intervals with homogeneous Neumann boundary conditions. In particular, we are interested in the speed of sharp interfaces approaching and annihilating each other. This process is known to be exponentially slow in the case of the classical Laplacian. Here we investigate how the width and speed of the interfaces change if we vary the exponent $α$ of the fractional Laplacian. For the associated model on the real-line we derive asymptotic formulas for the interface speed and time-to-collision in terms of $α$ and a scaling parameter $\varepsilon$. We use a numerical approach via a finite-element method based upon extending the fractional Laplacian to a cylinder in the upper-half plane, and compute the interface speed, time-to-collapse and interface width for $α\in(0.2,2]$. A comparison shows that the asymptotic formulas for the interface speed and time-to-collision give a good approximation for large intervals.

Metastable Speeds in the Fractional Allen-Cahn Equation

TL;DR

It is investigated how the width and speed of the interfaces change if the authors vary the exponent of the fractional Laplacian, and asymptotic formulas for the interface speed and time-to-collision give a good approximation for large intervals.

Abstract

We study numerically the one-dimensional Allen-Cahn equation with the spectral fractional Laplacian on intervals with homogeneous Neumann boundary conditions. In particular, we are interested in the speed of sharp interfaces approaching and annihilating each other. This process is known to be exponentially slow in the case of the classical Laplacian. Here we investigate how the width and speed of the interfaces change if we vary the exponent of the fractional Laplacian. For the associated model on the real-line we derive asymptotic formulas for the interface speed and time-to-collision in terms of and a scaling parameter . We use a numerical approach via a finite-element method based upon extending the fractional Laplacian to a cylinder in the upper-half plane, and compute the interface speed, time-to-collapse and interface width for . A comparison shows that the asymptotic formulas for the interface speed and time-to-collision give a good approximation for large intervals.

Paper Structure

This paper contains 16 sections, 4 theorems, 45 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Asymptotic analysis
  3. The Classical Case
  4. Analysis of fractional Allen-Cahn equations
  5. Analysis of fractional Allen-Cahn equations on $\mathbb{R}$
  6. Numerical Methods
  7. Detecting an interface and measuring speeds
  8. Model problem
  9. Comparison of the two methods
  10. Results on Metastability
  11. Interface speed $s=s(\varepsilon,\alpha)$
  12. Time--to--collapse $t_{col}$
  13. Interface width $w=w(\varepsilon,\alpha)$
  14. Summary and Outlook
  15. Dependence on the size of the domain
  16. ...and 1 more sections

Key Result

Theorem 2.1

Suppose $\alpha\in[1,2]$. Let where $\mathop{\mathrm{sgn}}\nolimits$ is the sign function and $(x_i(t))_{i=1,\ldots,2K}$ is the solution to ODE:centers. Then, for every $\varepsilon>0$ there exists a unique solution $u_\varepsilon$ of eq:AC:rescaled. Furthermore, as $\varepsilon\to0^+$, the solution $u_\varepsilon$ exhibits the for any $(t,x)\in[0,T_C)\times\mathbb{R}$.

Figures (10)

  • Figure 1: Evolution of solution for the classical Allen-Cahn equation \ref{['eq:AC']} for \ref{['eq:TL:simple']} with $L=5$, $\varepsilon=0.3$ and $\alpha=2$.
  • Figure 2: Example for the detected interfaces and roots.
  • Figure 3: Comparison of two numerical methods: hp--method vs. spectral--method. Different lines represent different values of $\alpha \in (0.2,1.9)$.
  • Figure 4: Comparing results on intervals $[-L,L]$ with $L=10,20,40$ and $\varepsilon=0.1$.
  • Figure 5: Evolution of solution for Allen-Cahn equation \ref{['eq:AC']} with well--prepared initial data \ref{['eq:TL:simple']} with $L=10$, $\varepsilon=0.01$ and $\alpha=0.9$.
  • ...and 5 more figures

Theorems & Definitions (6)

  • Theorem 2.1: cf. PaVa:2015
  • proof : Sketch of proof
  • Remark 2.2
  • Theorem 2.3: cf. PaVa:2015
  • Theorem 2.4: PaVa:2015
  • Theorem 2.5: cf. PaVa:2016