This paper is concerned with the numerical simulations for the dynamics of the time-fractional molecular beam epitaxy models. The variable-step Crank–Nicolson-type schemes are proposed and analyzed for model with and without slope selection respectively. By using the discrete gradient structures of discrete fractional derivative, the numerical schemes preserve discrete variational energy dissipation law on general meshes unconditionally. These are generalizations of classical energy dissipation laws and are asymptotically compatible with the classical MBE model. Numerical examples with an adaptive time-stepping strategy are provided to show the effectiveness of our schemes.

本文关注时间分数分子束外延模型动力学的数值模拟。分别针对有坡度选择和无坡度选择的模型提出并分析了变步长Crank-Nicolson型方案。数值格式利用离散分数导数的离散梯度结构，无条件地保留了一般网格上的离散变分能量耗散规律。这些是经典能量耗散定律的推广，并且与经典 MBE 模型渐近兼容。提供了具有自适应时间步长策略的数值示例以显示我们方案的有效性。