A Second-Order Imex-Rk Approach for Energy-Stable Phase Field Crystal Simulations

This paper presents a second-order, globally energy-stable IMEX-RK scheme for solving this formula numerically in the context of a particular field of application, namely, the phase field crystal (PFC) equation that has found extensive application in the field of material science to capture the atomic motions at the limit. By choosing an implicit and explicit scheme, in which there are more sub-step calculations within certain time steps, it also improves the solution of the stiffness issue and the nonlinearity within the PFC equation directly as implemented in the current study. By undertaking such a stability analysis, the scheme establishes that it supports energy stability and thus maintains the ratio of energy to the fluctuating inputs after some time. Numerical simulations confirm the effectiveness of the IMEX-RK2 method and demonstrate the method’s advantage over other numerical methods like the Runge-Kutta, Euler’s, and fully implicit methods. The result highlights the capability of the scheme for successful emulation of phase transition and material behaviours such as the SDKFILM and energy evolutions in solutions from the visualisation. This work can be considered useful for enhancing phase-field modelling using numerical methods and provides a wide range of opportunities for researchers in computational mathematics and materials science; it also paves the way for future studies on the influence of the presented scheme and more advanced ideas, including adaptive time-stepping schemes and further dimensions.