Mathematical Institute, University of Oxford, Oxford OX1 3LB, United Kingdom.
Proc Natl Acad Sci U S A. 2013 Jun 4;110(23):9209-14. doi: 10.1073/pnas.1221408110. Epub 2013 May 20.
The study of reaction-diffusion processes is much more complicated on general curved surfaces than on standard Cartesian coordinate spaces. Here we show how to formulate and solve systems of reaction-diffusion equations on surfaces in an extremely simple way, using only the standard Cartesian form of differential operators, and a discrete unorganized point set to represent the surface. Our method decouples surface geometry from the underlying differential operators. As a consequence, it becomes possible to formulate and solve rather general reaction-diffusion equations on general surfaces without having to consider the complexities of differential geometry or sophisticated numerical analysis. To illustrate the generality of the method, computations for surface diffusion, pattern formation, excitable media, and bulk-surface coupling are provided for a variety of complex point cloud surfaces.
反应-扩散过程的研究在一般曲面上比在标准笛卡尔坐标空间中要复杂得多。在这里，我们展示了如何以一种极其简单的方式在表面上制定和求解反应-扩散方程系统，仅使用标准笛卡尔形式的微分算子，以及一个离散的无组织点集来表示表面。我们的方法将表面几何形状与基础微分算子分离。因此，有可能在不考虑微分几何或复杂数值分析的复杂性的情况下，在一般表面上制定和求解相当一般的反应-扩散方程。为了说明该方法的通用性，针对各种复杂的点云表面，提供了表面扩散、模式形成、激发介质和体-面耦合的计算。
