功能化 Cahn-Hilliard 梯度流中形态复杂性的基准计算
Benchmark Computation of Morphological Complexity in the Functionalized Cahn-Hilliard Gradient Flow
AI总结：
针对功能化 Cahn-Hilliard 能量的奇异性，本文构建形态复杂性基准，并比较结合 Fourier 拟谱离散的 BDF2 全隐式、IMEX、SAV 与 ETD 格式及预处理效果。
摘要：
溶剂中两亲分子的自洽平均场理论模型经约化后，可得到一类奇异的功能化 Cahn-Hilliard 能量。我们对这些能量进行修改，通过缓和奇异性来稳定梯度流的计算，并开发了一系列基准问题，以模拟实验中观察到的“形态复杂性”。这些基准研究两亲物质向界面的吸收速率与分散到达质量的最低能量机制之间微妙的平衡。其结果是响应呈现三分法：二维界面要么通过逆曲率的正则化运动而延长，要么经历珠化分岔，要么直接分裂为界面网络。我们评估了多种采用二阶 BDF2 型时间步进并结合 Fourier 拟谱空间离散的格式。BDF2 型格式要么基于带 PSD 非线性求解器的全隐式时间离散，要么基于 IMEX、SAV、ETD 方法。所有格式均使用固定的局部截断误差目标，并通过自适应时间步长来达到该误差目标。每种格式都需要适当的“预处理”以实现稳健性能，这可将效率提高数个数量级。
英文摘要：
Reductions of the self-consistent mean field theory model of amphiphilic molecules in solvent can lead to a singular family of functionalized Cahn-Hilliard energies. We modify these energies, mollifying the singularities to stabilize the computation of the gradient flows and develop a series of benchmark problems that emulate the "morphological complexity" observed in experiments. These benchmarks investigate the delicate balance between the rate of absorption of amphiphilic material onto an interface and a least energy mechanism to disperse the arriving mass. The result is a trichotomy of responses in which two-dimensional interfaces either lengthen by a regularized motion against curvature, undergo pearling bifurcations, or split directly into networks of interfaces. We evaluate a number of schemes that use second order BDF2-type time stepping coupled with Fourier pseudo-spectral spatial discretization. The BDF2-type schemes are either based on a fully implicit time discretization with a PSD nonlinear solver, or upon IMEX, SAV, ETD approaches. All schemes use a fixed local truncation error target with adaptive time-stepping to achieve the error target. Each scheme requires proper "preconditioning" to achieve robust performance that can enhance efficiency by several orders of magnitude.
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