A Second-Order, Global-in-Time Energy Stable Implicit-Explicit Runge–Kutta Scheme for the Phase Field Crystal Equation

Abstract

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.