相场晶体方程的一个高精度能量稳定数值格式

An high-accuracy energy stable numerical scheme for the phase field crystal equation

摘要:

针对具有周期边界条件的相场晶体方程, 本文提出一个具有能量稳定性的高精度数值格式. 该格式基于方程的能量泛函的结构, 在空间上采用Fourier拟谱逼近, 在时间上进行三阶精度的后向差分离散，并在格式中增加Douglas-Dupont正则项, 以保证格式的能量稳定性, 本文证明了数值解的存在唯一性及数值格式的能量稳定性. 数值算例验证了算法的高精度和稳定性.

Abstract:

In this paper, we propose an energy stable numerical scheme for the phase field crystal equation with periodic boundary condition. This scheme is based on the structure of the energy functional. A Fourier pesudo spectral approximation is applied in space as well as a third order backward differentiation scheme is applied in the temporal approximation for the equation. Meanwhile, a Douglas-Dupont type regularization term is added to ensure the modified energy stability. The unique solvability and energy stability of the scheme are established. Finally, some numerical examples are presented to confirm the robustness and accuracy of the scheme.