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

This paper develops a high-order, unconditionally energy dissipative IMEX Runge--Kutta framework for phase field crystal models that does not rely on Lipschitz assumptions, ensuring stability and accuracy for complex nonlinear PDEs.

Contribution

It introduces a novel framework for high-order IMEX Runge--Kutta methods that guarantees unconditional energy stability without Lipschitz constraints, applicable to various gradient flow models.

Findings

Proves uniform boundedness and unconditional stability of solutions.

Derives optimal high-order L-infinity error estimates.

Validates results with numerical experiments.

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.