SGER: Nonlinear Diffusion on a Sphere

Palffy-MuhorayDMS-0908470 软凝聚态物理中的许多问题都需要求解球上的非线性扩散方程。 例如，巨囊泡中脂质的相分离，液晶顺序的时间演化，以及决定囊泡形状的缺陷分布。 本计画的目标是发展一个简单且有效的方法来数值描述球体上的非线性扩散。 第一个具体的问题要考虑的是解决方案的卡恩-希利亚德方程的单位球。 研究人员使用真实的空间离散化数值求解Cahn-Hilliard方程和相关方程。 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建规则网格（如果点的数量为20），因此他们计划使用随机网格，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，出现了三个迷人的主题：Voronoi曲面细分的领域，以确定每个网格站点的最近邻居;稳定和有效的计划近似微分算子;和正则化网格。 本计画提供一种新的演算法来辨识球面上任意点之间的最近邻点。 这对于解决各种各样的问题都很有用，从世界各地军事基地的优势区域和燃料库的定位到模拟囊泡和相关生物物体的行为。 此外，它提供了洞察与随机网格上的微分算子的近似值，和正则化策略，以改善随机网格的数值稳定性。 它使有效的数值解的非线性扩散方程的一个球，从而使各种各样的软凝聚态物理问题的数值描述。 该项目的一个重要方面是让来自代表性不足的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.