论文标题
空间分数Cahn-Hilliard模型的可变步骤BDF2方法的收敛分析
Convergence analysis of variable steps BDF2 method for the space fractional Cahn-Hilliard model
论文作者
论文摘要
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 $H^{-α}$, $α\in(0,1)$.傅立叶伪谱法用于空间近似。所提出的方案在足够的时步比率的足够限制下以修改的离散能量的形式继承了能量耗散法。使用新近证明的离散嵌入类型的卷积不平等，涉及分数拉普拉斯式的新离散的嵌入式卷积不平等。此外，从理论上讲，质量保护和独特的解决性也可以保证。进行数值实验以显示各种界面宽度的精度和能量耗散。特别是，解决方案的多个时间尺度演变是在短期到长时间仿真中通过自适应时间稳定策略捕获的。
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 $H^{-α}$, $α\in(0,1)$. 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.
