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Global-in-time energy stability analysis for the exponential time differencing Runge-Kutta scheme for the phase field crystal equation

Xiao Li, Zhonghua Qiao, Cheng Wang, Nan Zheng

TL;DR

This work proves a global-in-time energy stability result for a second-order exponential time differencing Runge–Kutta (ETDRK2) scheme applied to the phase field crystal (PFC) equation, by coupling a priori energy control at the previous step with a single-step $H_h^2$ estimate to bound the next step in the discrete maximum norm. The authors decompose the scheme into a stabilized linear part and a nonlinear part, introduce spectral tools in Fourier space, and select a stabilization parameter $ ext{kappa}$ to guarantee energy dissipation for any final time. A priori bounds on the numerical solution, together with a careful induction argument, yield $E_h(u^{n+1}) leq E_h(u^n)$ for all $n$, establishing global-in-time energy stability with respect to the original discrete energy $E_h$. The methodology provides a framework potentially extensible to other gradient-flow models and RK-type schemes, enabling reliable long-time simulations of gradient dynamics such as Cahn–Hilliard and Allen–Cahn/Cahn–Hilliard-type systems.

Abstract

The global-in-time energy estimate is derived for the second-order accurate exponential time differencing Runge-Kutta (ETDRK2) numerical scheme to the phase field crystal (PFC) equation, a sixth-order parabolic equation modeling crystal evolution. To recover the value of stabilization constant, some local-in-time convergence analysis has been reported, and the energy stability becomes available over a fixed final time. In this work, we develop a global-in-time energy estimate for the ETDRK2 numerical scheme to the PFC equation by showing the energy dissipation property for any final time. An a priori assumption at the previous time step, combined with a single-step $H^2$ estimate of the numerical solution, is the key point in the analysis. Such an $H^2$ estimate recovers the maximum norm bound of the numerical solution at the next time step, and then the value of the stabilization parameter can be theoretically justified. This justification ensures the energy dissipation at the next time step, so that the mathematical induction can be effectively applied, by then the global-in-time energy estimate is accomplished. This paper represents the first effort to theoretically establish a global-in-time energy stability analysis for a second-order stabilized numerical scheme in terms of the original free energy functional. The presented methodology is expected to be available for many other Runge-Kutta numerical schemes to the gradient flow equations.

Global-in-time energy stability analysis for the exponential time differencing Runge-Kutta scheme for the phase field crystal equation

TL;DR

This work proves a global-in-time energy stability result for a second-order exponential time differencing Runge–Kutta (ETDRK2) scheme applied to the phase field crystal (PFC) equation, by coupling a priori energy control at the previous step with a single-step estimate to bound the next step in the discrete maximum norm. The authors decompose the scheme into a stabilized linear part and a nonlinear part, introduce spectral tools in Fourier space, and select a stabilization parameter to guarantee energy dissipation for any final time. A priori bounds on the numerical solution, together with a careful induction argument, yield for all , establishing global-in-time energy stability with respect to the original discrete energy . The methodology provides a framework potentially extensible to other gradient-flow models and RK-type schemes, enabling reliable long-time simulations of gradient dynamics such as Cahn–Hilliard and Allen–Cahn/Cahn–Hilliard-type systems.

Abstract

The global-in-time energy estimate is derived for the second-order accurate exponential time differencing Runge-Kutta (ETDRK2) numerical scheme to the phase field crystal (PFC) equation, a sixth-order parabolic equation modeling crystal evolution. To recover the value of stabilization constant, some local-in-time convergence analysis has been reported, and the energy stability becomes available over a fixed final time. In this work, we develop a global-in-time energy estimate for the ETDRK2 numerical scheme to the PFC equation by showing the energy dissipation property for any final time. An a priori assumption at the previous time step, combined with a single-step estimate of the numerical solution, is the key point in the analysis. Such an estimate recovers the maximum norm bound of the numerical solution at the next time step, and then the value of the stabilization parameter can be theoretically justified. This justification ensures the energy dissipation at the next time step, so that the mathematical induction can be effectively applied, by then the global-in-time energy estimate is accomplished. This paper represents the first effort to theoretically establish a global-in-time energy stability analysis for a second-order stabilized numerical scheme in terms of the original free energy functional. The presented methodology is expected to be available for many other Runge-Kutta numerical schemes to the gradient flow equations.
Paper Structure (14 sections, 7 theorems, 113 equations)

This paper contains 14 sections, 7 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. The numerical scheme and a few preliminary estimates
  3. The finite difference spatial discretization
  4. The numerical scheme
  5. Some related operators in the Fourier space
  6. A few preliminary estimates
  7. The global-in-time energy stability estimate
  8. A preliminary estimate for $\| \tilde{u}^{n+1} \|_{H_h^2}$
  9. A preliminary estimate for $\| u^{n+1} \|_{H_h^2}$
  10. Justification of the artificial parameter $\kappa$ and the a priori assumption \ref{['a priori-1']}
  11. A refined estimate for $\| u^{n+1} \|_{H_h^2}$ and $\| u^{n+1} \|_\infty$
  12. Concluding remarks
  13. Proof of Proposition \ref{['prop:prop 1']}
  14. Proof of Proposition \ref{['prop:prop 2']}

Key Result

Lemma 2.1

For any $\psi, \phi \in {\mathcal{C}}_{\rm per}$ and any $\vec{f}$, the following summation-by-parts formulas are valid:

Theorems & Definitions (11)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Theorem 3.1
  • Remark 3.2
  • ...and 1 more