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Long-time stability analysis of an explicit exponential Runge-Kutta scheme for Cahn-Hilliard equations

Jing Guo

TL;DR

This work analyzes a two-stage, second-order explicit exponential Runge–Kutta scheme (ERK2) for the Cahn–Hilliard equation using a Fourier spectral collocation discretization. The authors prove uniform-in-time bounds in discrete $H^1$ and $H^2$ norms and an $\ell^\infty$ bound via a discrete Sobolev embedding, leading to unconditional energy dissipation for the fully discrete scheme once a stabilization parameter is chosen appropriately. Building on these stability results, they derive an optimal-order $\ell^2$-norm convergence estimate. The framework relies on a careful combination of eta-function based representations, operator estimates, and a priori energy dissipation to remove the usual boundedness assumptions common in energy-stability analyses, with potential extension to higher-order exponential integrators and a broad class of phase-field models.

Abstract

In this paper, we present a comprehensive long-time stability analysis of a second-order explicit exponential Runge--Kutta (ERK2) method for the Cahn--Hilliard (CH) equation. By employing Fourier spectral collocation in space and a two-stage ERK2 scheme in time, we construct a fully discrete numerical method that preserves the original energy dissipation property. The uniform-in-time boundedness of the numerical solution is rigorously proven in the discrete $H^1$ and $H^2$ norms under a mild time-step condition, and an $\ell^\infty$ bound is derived via a discrete Sobolev embedding. These results remove the typical boundedness assumption required in previous energy-stability analyses, thereby establishing unconditional energy dissipation for the fully discrete scheme. Building on this uniform boundedness, we derive an optimal-order error estimate in the $\ell^2$ norm. The analytical framework developed herein is general and can be extended to higher-order exponential integrators for a broader class of phase-field models.

Long-time stability analysis of an explicit exponential Runge-Kutta scheme for Cahn-Hilliard equations

TL;DR

This work analyzes a two-stage, second-order explicit exponential Runge–Kutta scheme (ERK2) for the Cahn–Hilliard equation using a Fourier spectral collocation discretization. The authors prove uniform-in-time bounds in discrete and norms and an bound via a discrete Sobolev embedding, leading to unconditional energy dissipation for the fully discrete scheme once a stabilization parameter is chosen appropriately. Building on these stability results, they derive an optimal-order -norm convergence estimate. The framework relies on a careful combination of eta-function based representations, operator estimates, and a priori energy dissipation to remove the usual boundedness assumptions common in energy-stability analyses, with potential extension to higher-order exponential integrators and a broad class of phase-field models.

Abstract

In this paper, we present a comprehensive long-time stability analysis of a second-order explicit exponential Runge--Kutta (ERK2) method for the Cahn--Hilliard (CH) equation. By employing Fourier spectral collocation in space and a two-stage ERK2 scheme in time, we construct a fully discrete numerical method that preserves the original energy dissipation property. The uniform-in-time boundedness of the numerical solution is rigorously proven in the discrete and norms under a mild time-step condition, and an bound is derived via a discrete Sobolev embedding. These results remove the typical boundedness assumption required in previous energy-stability analyses, thereby establishing unconditional energy dissipation for the fully discrete scheme. Building on this uniform boundedness, we derive an optimal-order error estimate in the norm. The analytical framework developed herein is general and can be extended to higher-order exponential integrators for a broader class of phase-field models.

Paper Structure

This paper contains 15 sections, 9 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Full discrete ERK scheme with Fourier collocation discretization
  3. Review of the Fourier collocation method
  4. Spatial semi-discrete system
  5. Exponential Runge--Kutta method and the energy stability results
  6. The proof of Theorem \ref{['thm:long_time_stability']}: Long-time numerical stability analysis
  7. Some preliminary estimates and setup
  8. Preliminary estimate for
  9. Preliminary estimate for
  10. Preliminary estimate for
  11. Uniform-in-time estimate for
  12. Recovery of the a priori assumption
  13. Convergence analysis
  14. Conclusion
  15. Proof of Lemma \ref{['lem:perFun_prop']}

Key Result

Lemma 2.1

For any $\bm f, \bm g \in \mathcal{M}_N$, the following inequalities hold Here and in what follows, $C$ denotes a generic positive constant that is independent of the discretization parameter but may depend on the domain $\Omega$.

Theorems & Definitions (14)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5: Energy stability of the ERK2 scheme
  • proof
  • Theorem 2.6
  • Lemma 3.1
  • Lemma 3.2
  • ...and 4 more