Efficient, Decoupled, Second and Third-Order Implicit-Explicit Runge–Kutta Schemes with Original-Form Energy Stability for the Anisotropic Phase-Field Dendritic Growth Model

Abstract. We consider numerical approximations of the anisotropic phase-field dendritic crystal growth model, a highly nonlinear system comprising the anisotropic Allen–Cahn equation and the thermal equation, coupled through the latent heat effect. Despite various efforts to develop efficient numerical approaches, significant challenges remain in constructing high-order (beyond second-order) and fully decoupled numerical schemes with provable energy stability, particularly with respect to the energy form corresponding to the original system. To address these challenges, we have developed a novel, linear, high-order (second or third) fully decoupled implicit-explicit Runge–Kutta scheme. Our schemes preserve energy stability in their original form, with only minor Lipschitiz modifications to the nonlinear potentials, which differ fundamentally from approaches that reformulate the entire free energy via auxiliary variables. The key to achieving energy stability lies in utilizing stabilization technique and various coefficient matrices for different nonlinear terms. We propose specific requirements for these coefficient matrices and introduce a search strategy to identify appropriate schemes, rigorously proving their energy stability. Numerous 2D and 3D simulations have been performed to demonstrate the stability and accuracy of the proposed schemes, numerically.