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Stability and convergence analysis of unconditionally original energy dissipative implicit-explicit Runge--Kutta methods for the phase field crystal models without Lipschitz assumptions

Xiaoli Li, Kaiyi Niu, Jiang Yang

TL;DR

The paper tackles high-order, energy-stable time discretization for the phase field crystal (PFC) equation without assuming Lipschitz continuity in the nonlinear term. It develops a stabilization-based, arbitrarily high-order IMEX-RK framework and an auxiliary Lipschitz-truncated problem, proving unconditional energy dissipation and uniform $L^{\infty}$ bounds through Cauchy interlacing arguments. A detailed $L^{\infty}$-error analysis shows convergence of order $p$ under mild time-step constraints, with numerical experiments in 2D and 3D validating long-time dynamics such as hexagonal pattern formation and crystal growth. The methodology extends to a broad class of gradient flows and offers a practical, structure-preserving approach for simulating complex microstructure evolution over long times.

Abstract

The phase field crystal (PFC) method is an efficient technique for simulating the evolution of crystalline microstructures at atomistic length scales and diffusive time scales. Due to the high-order derivatives (sixth-order) and the strongly nonlinear term (locally Lipschitz), developing high-order stable schemes and establishing corresponding error estimates is particularly challenging. In this study, we first establish a general framework for high-order implicit-explicit (IMEX) Runge--Kutta methods that preserves the original energy dissipation for auxiliary models with globally Lipschitz truncations on the nonlinear term. By employing the Sobolev embedding theorem and Cauchy's interlace theorem, we demonstrate that the solutions of the auxiliary models are identical to the solutions of the original models without the globally Lipschitz property, provided that the free energy of the initial value is well-defined. Furthermore, we rigorously prove the uniform boundedness of the solution in the L-infinity norm and unconditional global-in-time stability. This allows for a straightforward framework to derive optimal arbitrarily high-order L-infinity error estimate without relying on the Lipschitz assumption. In particular, compared to existing literature, the argument for error estimation is presented in a much more simplified and elegant manner, without imposing any constraints on time-step size or mesh grid size. In fact, the reported framework, built upon the truncated auxiliary problem for the original model, can be directly extended to a wide range of gradient flows, including Allen--Cahn equations, nonlocal PFC models, and epitaxial thin film growth equations, providing unconditional energy dissipation without enforcing Lipschitz continuity. Finally, we present numerical examples to validate our analytical results and demonstrate the effectiveness of capturing long-time dynamics.

Stability and convergence analysis of unconditionally original energy dissipative implicit-explicit Runge--Kutta methods for the phase field crystal models without Lipschitz assumptions

TL;DR

The paper tackles high-order, energy-stable time discretization for the phase field crystal (PFC) equation without assuming Lipschitz continuity in the nonlinear term. It develops a stabilization-based, arbitrarily high-order IMEX-RK framework and an auxiliary Lipschitz-truncated problem, proving unconditional energy dissipation and uniform bounds through Cauchy interlacing arguments. A detailed -error analysis shows convergence of order under mild time-step constraints, with numerical experiments in 2D and 3D validating long-time dynamics such as hexagonal pattern formation and crystal growth. The methodology extends to a broad class of gradient flows and offers a practical, structure-preserving approach for simulating complex microstructure evolution over long times.

Abstract

The phase field crystal (PFC) method is an efficient technique for simulating the evolution of crystalline microstructures at atomistic length scales and diffusive time scales. Due to the high-order derivatives (sixth-order) and the strongly nonlinear term (locally Lipschitz), developing high-order stable schemes and establishing corresponding error estimates is particularly challenging. In this study, we first establish a general framework for high-order implicit-explicit (IMEX) Runge--Kutta methods that preserves the original energy dissipation for auxiliary models with globally Lipschitz truncations on the nonlinear term. By employing the Sobolev embedding theorem and Cauchy's interlace theorem, we demonstrate that the solutions of the auxiliary models are identical to the solutions of the original models without the globally Lipschitz property, provided that the free energy of the initial value is well-defined. Furthermore, we rigorously prove the uniform boundedness of the solution in the L-infinity norm and unconditional global-in-time stability. This allows for a straightforward framework to derive optimal arbitrarily high-order L-infinity error estimate without relying on the Lipschitz assumption. In particular, compared to existing literature, the argument for error estimation is presented in a much more simplified and elegant manner, without imposing any constraints on time-step size or mesh grid size. In fact, the reported framework, built upon the truncated auxiliary problem for the original model, can be directly extended to a wide range of gradient flows, including Allen--Cahn equations, nonlocal PFC models, and epitaxial thin film growth equations, providing unconditional energy dissipation without enforcing Lipschitz continuity. Finally, we present numerical examples to validate our analytical results and demonstrate the effectiveness of capturing long-time dynamics.
Paper Structure (14 sections, 5 theorems, 53 equations, 8 figures, 1 table)

This paper contains 14 sections, 5 theorems, 53 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Energy preserving IMEX-RK methods for PFC model with stabilization technique
  3. A stabilization technique
  4. Implicit-Explicit Runge--Kutta method
  5. Energy dissipation and uniform boundedness
  6. Error analysis for PFC model
  7. Numerical experiments
  8. Convergence test
  9. Energy stability test
  10. 2D phase transition behaviors
  11. 2D crystal growth simulation
  12. Effect of τ on pattern formation
  13. 3D phase transition behaviors
  14. Conclusion

Key Result

Lemma 1

Suppose that $u$, $v:\bar{\Omega}\rightarrow\mathbb{R}$ are functions satisfying $\int_{\Omega}u\;dx=\int_{\Omega}v\;dx$ and $\mathcal{E}(u)\leq\mathcal{E}(v)$, equipped with either the periodic boundary condition or the homogeneous Neumann boundary condition. Then $\left\|u \right\|_{L^\infty}$ can

Figures (8)

  • Figure 1: Evolution results of the original energy for the four-stage third-order IMEX-RK method \ref{['eq2.8']} with $\epsilon=0.025$, $a=0.001$, $\tau=0.5$, different stabilizers and time steps at $T=120$
  • Figure 2: The snapshots of $\phi(x,y,t)$ using the four-stage third-order IMEX-RK method \ref{['eq2.8']} with $\epsilon=0.025$, $a=0.001$ and $\tau=0.1$ at $t=100,400,600,800,1200,2000$, respectively.
  • Figure 3: Energy evolution result for the phase behaviors. The inserting figure (left) showing the evolution of phase transition at $t=400,800,1200,2000$ and evolution of the energies (right) with $\tau=0.1,1,5,20$.
  • Figure 4: The evolution of crystal growth using the four-stage third-order IMEX-RK method \ref{['eq2.8']} with $\epsilon=0.25$, $a=0.001$ at $t=0,100,200,400,600,1500$, respectively.
  • Figure 5: Energy evolution result for the crystal growth. The inserting figure is the evolution of crystal growth at $t=100,200,600,1500$.
  • ...and 3 more figures

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3: Cauchy's interlace theorem
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof