The Curvature-Augmented Closest Point Method with Vesicle Inextensibility Application

TL;DR

This paper introduces a curvature-augmented closest point method that improves accuracy and efficiency for surface PDEs, demonstrated through vesicle shape evolution simulations.

Contribution

It develops a curvature-augmented operator embedding that eliminates the need for side condition enforcement, enhancing the original Closest Point method.

Findings

Better accuracy in surface PDE solutions.

Reduced memory usage due to increased matrix sparsity.

Successful application to vesicle shape evolution.

Abstract

The Closest Point method, initially developed by Ruuth and Merriman, allows for the numerical solution of surface partial differential equations without the need for a parameterization of the surface itself. Surface quantities are embedded into the surrounding domain by assigning each value at a given spatial location to the corresponding value at the closest point on the surface. This embedding allows for surface derivatives to be replaced by their Cartesian counterparts (e.g. $\nabla_s = \nabla$). This equivalence is only valid on the surface, and thus, interpolation is used to enforce what is known as the side condition away from the surface. To improve upon the method, this work derives an operator embedding that incorporates curvature information, making it valid in a neighborhood of the surface. With this, direct enforcement of the side condition is no longer needed. Comparisons in $\mathbb{R}^2$ and $\mathbb{R}^3$ show that the resulting Curvature-Augmented Closest Point method has better accuracy and requires less memory, through increased matrix sparsity, than the Closest Point method, while maintaining similar matrix condition numbers. To demonstrate the utility of the method in a physical application, simulations of inextensible, bi-lipid vesicles evolving toward equilibrium shapes are also included.