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An efficient proximal-based approach for solving nonlocal Allen-Cahn equations

Olena Burkovska, Ilyas Mustapha

Abstract

In this work, we present an efficient approach for the spatial and temporal discretization of the nonlocal Allen-Cahn equation, which incorporates various double-well potentials and an integrable kernel, with a particular focus on a non-smooth obstacle potential. While nonlocal models offer enhanced flexibility for complex phenomena, they often lead to increased computational costs and there is a need to design efficient spatial and temporal discretization schemes, especially in the non-smooth setting. To address this, we propose first- and second-order energy-stable time-stepping schemes combined with the Fourier collocation approach for spatial discretization. We provide energy stability estimates for the developed time-stepping schemes. A key aspect to our approach involves a representation of a solution via proximal operators. This together with the spatial and temporal discretizations enables direct evaluation of the solution that can bypass the solution of nonlinear, non-smooth, and nonlocal system. This method significantly improves computational efficiency, especially in the case of non-smooth obstacle potentials, and facilitates rapid solution evaluations in both two and three dimensions. We provide several numerical experiments to illustrate the effectiveness of our approach.

An efficient proximal-based approach for solving nonlocal Allen-Cahn equations

Abstract

In this work, we present an efficient approach for the spatial and temporal discretization of the nonlocal Allen-Cahn equation, which incorporates various double-well potentials and an integrable kernel, with a particular focus on a non-smooth obstacle potential. While nonlocal models offer enhanced flexibility for complex phenomena, they often lead to increased computational costs and there is a need to design efficient spatial and temporal discretization schemes, especially in the non-smooth setting. To address this, we propose first- and second-order energy-stable time-stepping schemes combined with the Fourier collocation approach for spatial discretization. We provide energy stability estimates for the developed time-stepping schemes. A key aspect to our approach involves a representation of a solution via proximal operators. This together with the spatial and temporal discretizations enables direct evaluation of the solution that can bypass the solution of nonlinear, non-smooth, and nonlocal system. This method significantly improves computational efficiency, especially in the case of non-smooth obstacle potentials, and facilitates rapid solution evaluations in both two and three dimensions. We provide several numerical experiments to illustrate the effectiveness of our approach.
Paper Structure (17 sections, 4 theorems, 85 equations, 12 figures, 1 algorithm)

This paper contains 17 sections, 4 theorems, 85 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Representation formula
  3. Contribution and related works
  4. Temporal discretization
  5. Semi-implicit first order scheme
  6. Second-order scheme
  7. Implicit second-order scheme
  8. Explicit second order scheme
  9. Energy stability estimates
  10. Spatial discretization
  11. Numerical experiments
  12. Convergence rates in time
  13. Sharp interfaces
  14. Computational costs
  15. Dynamics prediction for 3D examples.
  16. ...and 2 more sections

Key Result

Proposition 2.1

For the solution $u^k\in\mathcal{K}$ of eq:first_order_tk the following energy stability estimates hold

Figures (12)

  • Figure 1.1: From left to right: Illustration of different potentials $F$, corresponding sub-differentials $\partial F$, and proximal operators (for $\lambda = 1$), and the solution of \ref{['eq:general_xi']} at $t=10$ ($c_F=1$, $\xi=0.001$, and $\theta=0.25$).
  • Figure 4.1: Convergence rates in time of the first order, second order explicit, and second order implicit schemes for two- and three-dimensional test cases in Example 1 for obstacle potential.
  • Figure 4.2: Convergence rates in time of the first and second order schemes in the settings of Example 1 for logarithmic potential with $\theta_c=0.2$ (first column), regular potential solved with proximal operator (second column), and regular potential solved with fixed-point iteration (third column).
  • Figure 4.3: Time evolution of the solution of model \ref{['eq:general']} in the settings of Example 2 with obstacle potential (top), regular potential (middle), and logarithmic potential (bottom); and the corresponding energies for fixed $\varepsilon=0.1$ and different values of $\delta=0.1,\ 0.16$, and $0.1999$ corresponding to $\xi=3.0,\ 0.5625$ and $\xi=0.001$, respectively. From left to right: $t=10$, $20$ and $250.$
  • Figure 4.4: Time evolution of the solution of model \ref{['eq:general']} in the settings of Example 2 with the obstacle potential; and the corresponding energy for fixed $\delta=0.2$ and different values of $\varepsilon=0.2,\ 0.15$, and $0.1001$ corresponding to $\xi=3.0,\ 1.25$ and $\xi=0.002$, respectively. From left to right: $t=10$, $20$ and $250.$
  • ...and 7 more figures

Theorems & Definitions (9)

  • Proposition 2.1: Energy stability for the first-order scheme
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3: Energy stability estimates for the second order scheme
  • proof
  • Proposition 2.4: Convergence of the fixed point method
  • proof
  • Remark 4.1