An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation, involving the fractional Laplacian, derived from a gradient flow in the negative order Sobolev space $H^{-\alpha}$, $\alpha\in(0,1)$. The Fourier pseudo-spectral method is applied for the spatial approximation. The proposed scheme inherits the energy dissipation law in the form of the modified discrete energy under the sufficient restriction of the time-step ratios. The convergence of the fully discrete scheme is rigorously provided utilizing the newly proved discrete embedding type convolution inequality dealing with the fractional Laplacian. Besides, the mass conservation and the unique solvability are also theoretically guaranteed. Numerical experiments are carried out to show the accuracy and the energy dissipation both for various interface widths. In particular, the multiple-time-scale evolution of the solution is captured by an adaptive time-stepping strategy in the short-to-long time simulation.
翻译:为解决空间分数卡赫恩-希利亚德方程式问题,建立了一个隐含的可变步骤 BDF2 方案,涉及分数拉普拉西亚方程式,该方块来自负顺序Sobolev空间的梯度流($H ⁇ -\alpha}$, $>, ALphain\in( 0. 1) 美元)。 Fourier伪光谱法用于空间近似。 拟议的方案继承了以经修改的离散能源为形式的能量消散法, 且时间段比率受到足够限制。 利用新证明的离散嵌入型同分数拉帕西亚方圆形的不平等, 严格地提供了完全离散的组合。 此外, 质量保护以及独特的溶解能力在理论上也得到保证。 进行了数值实验, 以显示各种界面宽度的精度和能量消散。 特别是, 解决方案的多时段演化是通过短期模拟的适应性时间步动战略来捕捉摸取出多种时间尺度的解决方案的演变。