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

TL;DR

This paper tackles the challenge of solving PDEs on moving surfaces by marrying a least-squares implicit RBF-CPM with a grid-based particle approach. The method enforces a constant-along-normal extension via an auxiliary equation and a least-squares solve, enabling stable, high-order RBF-FD discretizations on flexible embedding tubes, including tubes with holes. It further couples this framework with GBPM to handle surface motion, demonstrating second-order convergence on moving geometries and applying it to diffusion, advection-diffusion, reaction-diffusion, and Cahn–Hilliard problems. The proposed approach provides a robust, implementable tool for PDEs on evolving manifolds with strong potential for adaptive tubes and complex topologies in practical applications.

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.

Figures (11)

• Figure 1: An illustration of the main steps of the GBPM (from top left to bottom right). Active grid points (blue dots) are connected to their footpoints (red dots) with blue lines. The green lines correspond to the grid.
• Figure 2: The relative error of the heat equation on the unit circle at $t=1$ using different backward differentiation formulas and augmented polynomial degree in our RBF-FD calculations.
• Figure 3: The relative error of the heat equation on the unit circle for $1\%$ (left) and $5\%$ (right) point removal at $t=1$ using BDF2 and up to third degree augmented polynomials in the RBF-FD calculation.
• Figure 4: The relative error of the heat equation on the unit circle at $t=1$ using different backward differentiation formulas and augmented polynomial degree in our RBF-FD calculations.
• Figure 5: The relative error of the heat equation on the unit sphere for $1\%$ (left) and $5\%$ (right) point removal at $t=1$ using BDF2 and up to third degree augmented polynomials in the RBF-FD calculation.

Theorems & Definitions (1)

• Definition 1