<p>In this paper, we present a class of decoupled, high-order, low-stage semi-implicit deferred correction (DC) schemes to simulate the phase-field dendritic crystal growth model with anisotropy, which are constructed upon the second-order backward differentiation formula (BDF) scheme. The model comprises a strongly nonlinear system coupling the anisotropic Allen-Cahn model with the thermal diffusion model. In order to linearize nonlinear terms arising from the gradient-dependent anisotropic coefficient, we introduce linear stabilization terms into both the initial BDF prediction and the correction stages within the DC framework. This design yields a linear, decoupled, semi-implicit, and computationally efficient scheme, as each iteration requires solving only a small number of Laplace-type problems with constant coefficients. From a theoretical perspective, we establish a rigorous proof of energy stability for the coupled second-order linear implicit BDF scheme in the isotropic case. Finally, a series of two- and three-dimensional numerical experiments on anisotropic models are conducted to test the convergence rates, energy dissipation behavior, and computational efficiency of the developed decoupled semi-implicit algorithms.</p>

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Semi-implicit low-stage deferred correction methods for the anisotropic phase-field dendritic crystal growth model

  • Lin Yao,
  • Yinhua Xia,
  • Yan Xu

摘要

In this paper, we present a class of decoupled, high-order, low-stage semi-implicit deferred correction (DC) schemes to simulate the phase-field dendritic crystal growth model with anisotropy, which are constructed upon the second-order backward differentiation formula (BDF) scheme. The model comprises a strongly nonlinear system coupling the anisotropic Allen-Cahn model with the thermal diffusion model. In order to linearize nonlinear terms arising from the gradient-dependent anisotropic coefficient, we introduce linear stabilization terms into both the initial BDF prediction and the correction stages within the DC framework. This design yields a linear, decoupled, semi-implicit, and computationally efficient scheme, as each iteration requires solving only a small number of Laplace-type problems with constant coefficients. From a theoretical perspective, we establish a rigorous proof of energy stability for the coupled second-order linear implicit BDF scheme in the isotropic case. Finally, a series of two- and three-dimensional numerical experiments on anisotropic models are conducted to test the convergence rates, energy dissipation behavior, and computational efficiency of the developed decoupled semi-implicit algorithms.