Decoupled energy-stable Runge-Kutta schemes of arbitrary order for the anisotropic phase-field dendritic crystal growth model
Abstract
In this paper, we construct an arbitrary-order scheme for the anisotropic phase-field dendritic crystal growth model by introducing a time dependent auxiliary variable. By employing an algebraically stable Runge-Kutta method, the proposed scheme satisfies an unconditional discrete energy dissipation law. To reduce the computational cost, a matrix diagonalization technique is applied to the coupled elliptic system at each time step. This transforms the original system into independent elliptic equations with constant coefficients, which can be solved separately or in parallel. After the decoupling, the auxiliary variable is obtained from a uniquely solvable q× q algebraic system. For a fixed Fourier-Galerkin space, we further prove qth-order convergence in time for the scheme based on a q-stage Runge-Kutta method. Numerical experiments in two and three dimensions confirm the theoretical convergence rates and the discrete energy dissipation, and demonstrate the computational efficiency of the decoupled schemes. The effects of anisotropy, latent heat, orientation angle, and initial nuclei on the dendritic morphology are also investigated numerically.
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BibTeX
@misc{wang2026decoupled,
title = {{Decoupled energy-stable Runge-Kutta schemes of arbitrary order for the anisotropic phase-field dendritic crystal growth model}},
author = {Weiwen Wang and Shaoqin Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.25345},
eprint = {2606.25345},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.25345},
}
RIS
TY - UNPB
TI - Decoupled energy-stable Runge-Kutta schemes of arbitrary order for the anisotropic phase-field dendritic crystal growth model
AU - Weiwen Wang
AU - Shaoqin Zheng
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2606.25345
UR - https://arxiv.org/abs/2606.25345
AB - In this paper, we construct an arbitrary-order scheme for the anisotropic phase-field dendritic crystal growth model by introducing a time dependent auxiliary variable. By employing an algebraically stable Runge-Kutta method, the proposed scheme satisfies an unconditional discrete energy dissipation law. To reduce the computational cost, a matrix diagonalization technique is applied to the coupled elliptic system at each time step. This transforms the original system into independent elliptic equations with constant coefficients, which can be solved separately or in parallel. After the decoupling, the auxiliary variable is obtained from a uniquely solvable q× q algebraic system. For a fixed Fourier-Galerkin space, we further prove qth-order convergence in time for the scheme based on a q-stage Runge-Kutta method. Numerical experiments in two and three dimensions confirm the theoretical convergence rates and the discrete energy dissipation, and demonstrate the computational efficiency of the decoupled schemes. The effects of anisotropy, latent heat, orientation angle, and initial nuclei on the dendritic morphology are also investigated numerically.
ER -