In this paper, we present a class of high-order, large time-stepping, and delay-free stabilization schemes for the Allen-Cahn equation. First, we apply a Fourier pseudo-spectral method for spatial discretization, and then, we establish the \(l^{2}\)-bound of the semi-discrete system. Furthermore, by adopting a time-step-dependent stabilization technique and taking advantage of recursive approximation of the exponential functions, we propose a class of stabilization Runge-Kutta schemes that preserve \(l^2\)-bound for any time-step size. Finally, we eliminate the delayed convergence brought by stabilization via a relaxation technique. Consequently, the resulting up-to-fourth-order parametric relaxation integrating factor Runge-Kutta (pRIFRK) schemes preserve the \(l^{2}\)-boundedness unconditionally with suitably chosen stabilization parameters. We also prove that the first-order pRIFRK scheme is unconditionally dissipative, w.r.t. a modified energy function, and the temporal convergence in the \(l^{2}\)-norm is estimated with pth-order accuracy. Numerical experiments are carried out to demonstrate the high-order accuracy, structure-preserving properties, and performance of the proposed schemes.

在本文中，我们提出了一类用于 Allen-Cahn 方程的高阶、大时间步长和无延迟稳定方案。首先，我们应用傅里叶伪谱方法进行空间离散化，然后，我们建立半离散系统的\(l^{2}\)界。此外，通过采用时间步长相关的稳定技术并利用指数函数的递归逼近，我们提出了一类稳定龙格-库塔方案，该方案在任何时间步长下都保留\(l^2\)界限。最后，我们通过松弛技术消除了稳定带来的延迟收敛。因此，所得到的高达四阶参数松弛积分因子龙格-库塔 (pRIFRK) 方案通过适当选择的稳定参数无条件地保留\(l^{2}\)有界性。我们还证明一阶 pRIFRK 方案对于修改后的能量函数来说是无条件耗散的，并且\(l^{2}\)范数中的时间收敛性以p阶精度估计。数值实验证明了所提出方案的高阶精度、结构保持特性和性能。