SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2667-2697, December 2024.
Abstract. We develop a two-stage, second-order, global-in-time energy stable implicit-explicit Runge–Kutta (IMEX RK(2, 2)) scheme for the phase field crystal equation with an [math] time step constraint, and without the global Lipschitz assumption. A linear stabilization term is introduced to the system with Fourier pseudo-spectral spatial discretization, and a novel compact reformulation is devised by rewriting the IMEX RK(2, 2) scheme as an approximation to the variation-of-constants formula. Under the assumption that all stage solutions are a priori bounded in the [math] norm, we first demonstrate that the original energy obtained by this second-order scheme is nonincreasing for any time step with a sufficiently large stabilization parameter. To justify the a priori [math] bound assumption, we establish a uniform-in-time [math] estimate for all stage solutions, subject to an [math] time step constraint. This results in a uniform-in-time bound for all stage solutions through discrete Sobolev embedding from [math] to [math]. Consequently, we achieve an [math] stabilization parameter, ensuring global-in-time energy stability. Additionally, we provide an optimal rate convergence analysis and error estimate for the IMEX RK(2, 2) scheme in the [math] norm. The global-in-time energy stability represents a novel achievement for a two-stage, second-order accurate scheme for a gradient flow without the global Lipschitz assumption. Numerical experiments substantiate the second-order accuracy and energy stability of the proposed scheme.

中文翻译：

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相场晶体方程的二阶、全局时间能量稳定隐式显式 Runge-Kutta 方案


SIAM 数值分析杂志，第 62 卷，第 6 期，第 2667-2697 页，2024 年 12 月。

抽象。我们为相场晶体方程开发了一个两阶段、二阶、全局能量稳定的隐式显式 Runge-Kutta （IMEX RK（2， 2）） 方案，具有 [数学] 时间步长约束，没有全局 Lipschitz 假设。通过傅里叶伪谱空间离散化向系统引入线性稳定项，并通过重写 IMEX RK（2， 2） 方案作为常数变化公式的近似值来设计一种新的紧凑重新公式。假设所有阶段解在 [数学] 范数中都是先验界的，我们首先证明通过这种二阶方案获得的原始能量对于具有足够大的稳定参数的任何时间步都是不增加的。为了证明先验 [数学] 有界假设的合理性，我们为所有阶段解建立了一个均匀的时间 [数学] 估计，但要受到 [数学] 时间步长约束。这导致通过从 [math] 到 [math] 的离散 Sobolev 嵌入为所有阶段解提供均匀的时间界限。因此，我们实现了一个 [数学] 稳定参数，确保了全局能量稳定性。此外，我们还为 [math] 范数中的 IMEX RK（2， 2） 方案提供了最佳速率收敛分析和误差估计。全局时间能量稳定性代表了没有全局 Lipschitz 假设的梯度流的两阶段二阶精确方案的新成就。数值实验证实了所提方案的二阶精度和能量稳定性。