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Explicit adaptive time stepping for the Cahn-Hilliard equation by exponential Krylov subspace and Chebyshev polynomial methods

Mike A. Botchev

TL;DR

This work tackles the time integration of the stiff Cahn–Hilliard equation by proposing two explicit schemes: EE2, an exponential Euler method realized via restarted Krylov subspaces, and LIM, a Chebyshev-based local-iteration approach. Both methods are equipped with adaptive time stepping, and EE2 is analyzed for nonstiff accuracy order 2 while LIM relies on Chebyshev stability properties. Through extensive 2D CH experiments, the paper demonstrates that EE2 can outperform LIM in efficiency (fewer matvecs) at comparable accuracy, especially when adaptivity is enabled; Eyre convex splitting can stabilize the methods but may degrade accuracy. The results highlight the potential of explicit exponential-time stepping for large-scale CH problems on parallel/hybrid architectures and identify directions for improving robustness and extending to higher orders.

Abstract

The Cahn-Hilliard equation has been widely employed within various mathematical models in physics, chemistry and engineering. Explicit stabilized time stepping methods can be attractive for time integration of the Cahn-Hilliard equation, especially on parallel and hybrid supercomputers. In this paper, we propose an exponential time integration method for the Cahn-Hilliard equation and describe its efficient Krylov subspace based implementation. We compare the method to a Chebyshev polynomial local iteration modified (LIM) time stepping scheme. Both methods are explicit (i.e., do not involve linear system solution) and tested with both constant and adaptively chosen time steps.

Explicit adaptive time stepping for the Cahn-Hilliard equation by exponential Krylov subspace and Chebyshev polynomial methods

TL;DR

This work tackles the time integration of the stiff Cahn–Hilliard equation by proposing two explicit schemes: EE2, an exponential Euler method realized via restarted Krylov subspaces, and LIM, a Chebyshev-based local-iteration approach. Both methods are equipped with adaptive time stepping, and EE2 is analyzed for nonstiff accuracy order 2 while LIM relies on Chebyshev stability properties. Through extensive 2D CH experiments, the paper demonstrates that EE2 can outperform LIM in efficiency (fewer matvecs) at comparable accuracy, especially when adaptivity is enabled; Eyre convex splitting can stabilize the methods but may degrade accuracy. The results highlight the potential of explicit exponential-time stepping for large-scale CH problems on parallel/hybrid architectures and identify directions for improving robustness and extending to higher orders.

Abstract

The Cahn-Hilliard equation has been widely employed within various mathematical models in physics, chemistry and engineering. Explicit stabilized time stepping methods can be attractive for time integration of the Cahn-Hilliard equation, especially on parallel and hybrid supercomputers. In this paper, we propose an exponential time integration method for the Cahn-Hilliard equation and describe its efficient Krylov subspace based implementation. We compare the method to a Chebyshev polynomial local iteration modified (LIM) time stepping scheme. Both methods are explicit (i.e., do not involve linear system solution) and tested with both constant and adaptively chosen time steps.

Paper Structure

This paper contains 19 sections, 3 theorems, 44 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem setting
  3. LIM (local iteration modified) time integration scheme
  4. Description of the LIM scheme
  5. The LIM scheme combined with the Eyre convex splitting
  6. Adaptive time step size selection in the LIM scheme
  7. EE2 (exponential Euler order 2) time integration scheme
  8. EE2 scheme, Krylov subspace residual and restarting
  9. Accuracy order of EE2
  10. The EE2 scheme combined with the Eyre convex splitting
  11. Adaptive time step selection in the EE2 scheme
  12. Numerical experiments
  13. Test setting
  14. The LIM and EE2 schemes with a constant time step size
  15. LIM and EE2 schemes with the Eyre splitting
  16. ...and 4 more sections

Key Result

Proposition 1

Let $A\in\mathbb{R}^{N\times N}$ defined in IVP be a symmetric positive semidefinite matrix. Then the matrix $\widehat{A}_n$ defined in A_g is diagonalizable, has real eigenvalues and the number of its negative (positive) eigenvalues does not exceed the number of negative (positive) eigenvalues of t

Figures (12)

  • Figure 1: A time step of the LIM scheme based on the linearized implicit Euler scheme \ref{['EBlin']}, with $\widehat{A}_n$ and $\widehat{g}^n$ defined by either \ref{['A_g']} or \ref{['A_g_cs']}
  • Figure 2: The LIM adaptive time stepping scheme
  • Figure 3: Krylov subspace method to make an EE2 time step by solving \ref{['IVPlin']}, with $\widehat{A}_n$ and $\widehat{g}^n$ defined by either \ref{['A_g']} or \ref{['A_g_cs']}. Here $H_{j,j}\in\mathbb{R}^{j\times j}$ is formed by the first $j$ rows and columns of $H_{m,m}$ and $V_j\in\mathbb{R}^{N\times j}$ by the first $j$ columns of $V_m$.
  • Figure 4: The EE2 adaptive time stepping scheme
  • Figure 5: Error convergence plots for the LIM (left) and the EE2 (right) schemes run with constant time step size $\tau$ on the $128\times 128$ space grid. The plots are produced with the data from Table \ref{['t:dt_const']}.
  • ...and 7 more figures

Theorems & Definitions (8)

  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 2
  • Proposition 3
  • proof