A SECOND-ORDER, GLOBAL-IN-TIME ENERGY STABLE IMPLICIT-EXPLICIT RUNGE-KUTTA SCHEME FOR THE PHASE FIELD CRYSTAL EQUATION

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 O(1) 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 l(infinity) 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 l(infinity) bound assumption, we establish a uniform-in-time H-N(2) estimate for all stage solutions, subject to an O(1) time step constraint. This results in a uniform-in-time bound for all stage solutions through discrete Sobolev embedding from H-N(2) to l(infinity). Consequently, we achieve an O(1) 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 l(infinity) (0, T; H-N(2)) boolean AND l(2)(0, T; H-N(5)) 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.