论文标题
具有对数电势的官能化的Cahn-Hilliard方程的独特可解决，具有阳性和无条件能量稳定的数值方案
A Uniquely Solvable, Positivity-Preserving and Unconditionally Energy Stable Numerical Scheme for the Functionalized Cahn-Hilliard Equation with Logarithmic Potential
论文作者
论文摘要
我们提出并分析具有对数的Flory-Huggins电位的功能化Cahn-Hilliard（FCH）方程的一阶有限差方案。半平式数值方案的设计基于FCH自由能的合适凸孔分解。我们证明了数值算法的独特可溶性，并验证其无条件的能量稳定性，而无需限制时间步长。由于纯状态$ \ pm 1 $附近的弗洛里 - 追随者潜力中对数部分的奇异性质，我们在理论级别建立了相位函数的所谓阳性保留属性。结果，数值解决方案将永远不会在点含义上达到单数值$ \ pm 1 $，并且在每个时间步骤中都可以很好地定义完全离散的方案。接下来，我们提出详细的最佳速率收敛分析，并在线性细化要求下$ l^{\ infty}（0，t; t; l_h^2）\ cap l^2（0，t; h^3_h）$以$ l^{\ infty}（0，t; l_h^2）$ ter误差估计。为了实现目标，利用较高的渐近扩展（直至二阶时间和空间精度）基于傅立叶投影来控制数值方案的离散最大解决方案最大规范。我们表明，如果将连续问题的精确解决方案严格与纯状态$ \ pm 1 $分开，则可以通过正距离的正距离将数值解决方案从$ \ pm 1 $中远离，该距离相对于时间步的大小和网格均匀。最后，提出了一些数值实验。进行收敛测试以证明所提出的数值方案的准确性和鲁棒性。在数值模拟中观察到珍珠分叉，弯曲的不稳定性和旋转分解。
We propose and analyze a first-order finite difference scheme for the functionalized Cahn-Hilliard (FCH) equation with a logarithmic Flory-Huggins potential. The semi-implicit numerical scheme is designed based on a suitable convex-concave decomposition of the FCH free energy. We prove unique solvability of the numerical algorithm and verify its unconditional energy stability without any restriction on the time step size. Thanks to the singular nature of the logarithmic part in the Flory-Huggins potential near the pure states $\pm 1$, we establish the so-called positivity-preserving property for the phase function at a theoretic level. As a consequence, the numerical solutions will never reach the singular values $\pm 1$ in the point-wise sense and the fully discrete scheme is well defined at each time step. Next, we present a detailed optimal rate convergence analysis and derive error estimates in $l^{\infty}(0,T;L_h^2)\cap l^2(0,T;H^3_h)$ under a linear refinement requirement $Δt\leq C_1 h$. To achieve the goal, a higher order asymptotic expansion (up to the second order temporal and spatial accuracy) based on the Fourier projection is utilized to control the discrete maximum norm of solutions to the numerical scheme. We show that if the exact solution to the continuous problem is strictly separated from the pure states $\pm 1$, then the numerical solutions can be kept away from $\pm 1$ by a positive distance that is uniform with respect to the size of the time step and the grid. Finally, a few numerical experiments are presented. Convergence test is performed to demonstrate the accuracy and robustness of the proposed numerical scheme. Pearling bifurcation, meandering instability and spinodal decomposition are observed in the numerical simulations.
