Decoupled energy-stable Runge-Kutta schemes of arbitrary order for the anisotropic phase-field dendritic crystal growth model

Abstract

This paper introduces a novel and general framework for constructing arbitrary-order, energy-stable, and decoupled schemes for the anisotropic phase-field dendritic crystal growth model. The proposed framework combines a time-dependent auxiliary variable, an algebraically stable Runge-Kutta method, and a matrix diagonalization technique. This unique combination yields a scheme that is proven to be unconditionally energy-dissipative while requiring only the solution of independent, constant-coefficient elliptic equations at each time step, which can be solved in parallel. The auxiliary variable is recovered from a uniquely solvable $q\times q$ algebraic system. We provide a rigorous proof of $q$th-order convergence in time for a $q$-stage Runge-Kutta method within a fixed Fourier-Galerkin space. Numerical experiments in 2D and 3D not only confirm the theoretical convergence and energy stability but also demonstrate the efficiency of the decoupled approach and provide a detailed investigation of key parameters' effects on dendritic morphology.

arXiv comment36 pages, 119 figures, 5 tables