SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1052-A1070, January 2020.
We present a general purpose method for solving partial differential equations on a closed surface, based on a technique for discretizing the surface introduced by Wenjun Ying and Wei-Cheng Wang [J. Comput. Phys., 252 (2013), pp. 606--624] which uses projections on coordinate planes. Assuming it is given as a level set, the surface is represented by a set of points at which it intersects the intervals between grid points in a three-dimensional grid. They are designated as primary or secondary. Discrete functions on the surface have independent values at primary points, with values at secondary points determined by an equilibration process. Each primary point and its neighbors have projections to regular grid points in a coordinate plane where the equilibration is done and finite differences are computed. The solution of a p.d.e. can be reduced to standard methods on Cartesian grids in the coordinate planes, with the equilibration allowing seamless transition from one system to another. We observe second order accuracy in examples with a variety of equations, including surface diffusion determined by the Laplace--Beltrami operator and the shallow water equations on a sphere.

中文翻译：

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用平面笛卡尔网格求解封闭表面上的偏微分方程

SIAM科学计算杂志，第42卷，第2期，第A1052-A1070页，2020年1月。
我们提出了一种通用的方法来求解封闭表面上的偏微分方程，该方法基于一种由问文俊和王伟成介绍的离散化表面的技术。计算 Phys。，252（2013），pp。606--624]，它使用了坐标平面上的投影。假设将其指定为水平集，则曲面由一组点表示，该点与三维网格中网格点之间的间隔相交。它们被指定为主要或次要的。表面上的离散函数在主要点上具有独立的值，而在次要点上的值由平衡过程确定。每个主点及其邻居在一个坐标平面中都有投影到规则网格点的位置，平衡完成并计算了有限的差。PDE的解决方案 可以在坐标平面上简化为笛卡尔网格上的标准方法，并且可以实现从一个系统到另一个系统的无缝过渡。我们在具有各种方程式的示例中观察到二阶精度，包括由Laplace-Beltrami算子确定的表面扩散和球体上的浅水方程式。