High-order \(L^{2}\)-bound-preserving Fourier pseudo-spectral schemes for the Allen-Cahn equation

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 \(l^{2}\) $l^2$ -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 \(l^2\) $l^2$ -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 \(l^{2}\) $l^2$ -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 \(l^{2}\) $l^2$ -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.