# A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

## TL;DR

This paper introduces a least-squares implicit RBF-FD closest point method for solving PDEs on moving surfaces, enhancing flexibility and accuracy in complex geometries and dynamic surface problems.

## Contribution

It develops a novel implicit formulation of the RBF-FD closest point method that allows flexible grid choices and integration with particle methods for moving surfaces.

## Key findings

- Demonstrates numerical convergence of the proposed method.
- Successfully applies to advection-diffusion and Cahn-Hilliard equations.
- Shows improved handling of complex and dynamic geometries.

## Abstract

The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection-diffusion equations and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented.