SGER: Nonlinear Diffusion on a Sphere

软凝聚态物理中的许多问题都需要求解球面上的非线性扩散方程。例如巨囊泡中脂质的相分离，液晶序的时间演化，以及决定囊泡形状的缺陷分布。本课题的目标是开发一种简单有效的方法来描述球面上的非线性扩散。要考虑的第一个具体问题是单位球上Cahn-Hilliard方程的解。研究人员采用实空间离散方法，对Cahn-Hilliard方程及相关方程进行了数值求解。数值模拟对底层晶格的细节很敏感。由于不可能在球体上构造规则晶格（如果点的数量为20），他们计划使用随机晶格，通过在球体表面随机放置点来获得。在设计实现这一目标的策略时，出现了三个引人入胜的主题：球体的Voronoi镶嵌，以识别每个晶格点的最近邻居；稳定高效的近似微分算子格式并使网格规范化。本文提出了一种新的球面随机点之间的最近邻识别算法。这对于解决各种各样的问题很有用，从世界各地军事基地的主导区域和燃料库的定位到对囊泡和相关生物物体的行为建模。此外，它还提供了与随机网格上的微分算子近似值相关的数值稳定性的见解，以及改进随机网格的正则化策略。它使球面上非线性扩散方程的有效数值解成为可能，从而使软凝聚态物理中各种问题的数值描述成为可能。该项目的一个重要方面是让来自代表性不足群体的学生和博士后积极参与这项令人兴奋的研究。

Palffy-MuhorayDMS-0908470 A wide variety of problems in soft condensed matter physics require the solution of a nonlinear diffusion equation on a sphere. Examples are the phase separation of lipids in giant vesicles, the time evolution of liquid crystalline order, and the distribution of defects that determine the shapes of vesicles. The goal of this project is to develop a simple and effective scheme for the numerical description of nonlinear diffusion on a sphere. The first specific problem to be considered is the solution of the Cahn-Hilliard equation on the unit sphere. The investigators solve the Cahn-Hilliard and related equations numerically, using real space discretization. Numerical simulations are sensitive to details of the underlying lattice. Because it is not possible to construct a regular lattice on a sphere (if the number of points 20), they plan to use a random lattice, obtained by placing points randomly on the surface of the sphere. In devising a strategy towards this goal, three fascinating topics emerge: Voronoi tessellation of the sphere to identity the nearest neighbors of each lattice site; stable and efficient scheme to approximate differential operators; and regularizing the grid. This project provides a new algorithm to identify nearest neighbors among random points on a sphere. This is useful for solving a wide variety of problems, ranging from dominance regions of military bases around the world and the locating of fuel depots to modeling the behavior of vesicles and related biological objects. In addition, it gives insights into numerical stability associated with the approximants of differential operators on random lattices, and regularization strategies for improving random grids. It enables the effective numerical solution of nonlinear diffusion equations on a sphere, and thereby makes possible numerical descriptions of a wide variety of problems in soft condensed matter physics. An important aspect of the project is to get students and postdocs from underrepresented groups actively involved in this exciting research.