Algorithms for continuum processes on complex, moving surfaces

连续介质过程的偏微分方程(PDE)模型遍及应用科学和自然科学。对于一维、二维或三维空间中的许多重要的偏微分方程类，人们已经做出了很大的努力来发展数值方法。然而，有许多问题涉及曲面上的微分方程式。曲面上偏微分方程组的数值方法比在标准笛卡尔坐标空间R2或R3中复杂得多。求解这类偏微分方程组的一类流行的方法是嵌入方法。在最近的研究中，我们引入了最近点方法，这是一种将曲面几何与底层微分算子解耦的嵌入方法。这项研究是导致2011年Germund Dahquist奖的工作的一部分。 最近点法利用嵌入空间中的标准数值偏微分方程组方法，精确地求解光滑、静止曲面上的一般偏微分方程组。然而，许多实际问题涉及到非光滑、移动的表面。在这项资助中，将对最近点方法进行一些几何改进：(1)研究具有褶皱、连接和其他非光滑特征的曲面上的一般偏微分方程组的算法；(2)研究复杂、不断变化的曲面上的偏微分方程组的稳健逼近方法；(3)研究点云曲面上的一般偏微分方程的精确方法。除了这些几何改进外，还将使用区域分解和优化的时间步长方法来开发效率增强。还将进行分析研究，为该方法的收敛提供新的见解，特别是在其表现出人意料地准确的情况下。在整个过程中，高素质的人员将参与数学软件的开发、数值算法的分析和数值方法的设计，以求解曲面上的偏微分方程组。这为学术机构和技术相关行业的广泛职业提供了宝贵的经验。 拟议的研究将寻求准确和高效的算法和软件，但从简单的意义上说，它们在充分利用现有的3D技术和软件的同时，尽可能统一地计算不同连续介质模型的解。预计在这项资助下获得的方法将使研究人员和最终用户能够以极高的精度研究复杂运动表面上的一般连续统过程的数值现实模型。

Partial differential equation (PDE) models for continuum processes arise throughout the applied and natural sciences. There has been a great effort made to develop numerical methods for many important classes of PDEs in one, two or three spatial dimensions. However, a remarkable variety of problems involve differential equations on curved surfaces. Numerical methods for PDEs on curved surfaces are much more complicated than in the standard Cartesian coordinate spaces R2 or R3. A popular class of methods for solving such PDEs are the embedding methods. In recent research, we introduced the closest point method, which is an embedding method that decouples surface geometry from the underlying differential operators. This research is part of the work that led to the 2011 Germund Dahquist Prize. The closest point method solves rather general PDEs on smooth, stationary surfaces in an accurate manner, using standard numerical PDE methods in the embedding space. However, many practical problems involve nonsmooth, moving surfaces. In this grant, a number of geometric enhancements to the closest point method will be introduced: (1) Research will be conducted on algorithms for general PDEs on surfaces with folds, junctions and other nonsmooth features, (2) robust methods for approximating PDEs on complex, evolving surfaces will be developed, (3) research on accurate methods for general PDEs on point cloud surfaces will also be carried out. Alongside these geometric improvements, efficiency enhancements will be developed using domain decomposition and optimized time-stepping methods. Analytical research will also be conducted to provide new insight into the convergence of the method, especially in situations where its performance is unexpectedly accurate. Throughout, highly qualified personnel will participate in the development of mathematical software, the analysis of numerical algorithms, and the design of numerical methods to solve PDEs on surfaces. This provides valuable experience for a broad range of careers in academic institutions and technology-related industries. The proposed research will seek algorithms and software that are accurate and efficient, yet are simple in the sense that they compute solutions to different continuum models as uniformly as possible while making strong use of existing techniques and software in 3D. It is anticipated that the methods obtained under this grant will enable researchers and end-users to numerically investigate realistic models of general continuum processes on complex moving surfaces with great accuracy.