# Solving Partial Differential Equations on Closed Surfaces with Planar Cartesian Grids

## TL;DR

This paper introduces a versatile method for solving PDEs on closed surfaces by discretizing the surface using projections onto coordinate planes, enabling standard Cartesian grid techniques to be applied.

## Contribution

It develops a novel surface discretization approach based on projections and equilibration, allowing PDE solutions on surfaces with second order accuracy.

## Key findings

- Achieved second order accuracy in surface diffusion problems.
- Successfully applied method to shallow water equations on a sphere.
- Demonstrated versatility across various PDEs on closed surfaces.

## Abstract

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.