A Robust Hyperviscosity Formulation for Stable RBF-FD Discretizations of Advection-Diffusion-Reaction Equations on Manifolds

TL;DR

This paper introduces a new hyperviscosity approach for stabilizing RBF-FD discretizations of advection-diffusion-reaction equations on manifolds, achieving high-order accuracy and stability even on point cloud representations.

Contribution

The paper develops an automatic hyperviscosity formulation for RBF-FD on manifolds, overcoming previous limitations and enabling stable, high-order accurate discretizations on point clouds.

Findings

High-order convergence demonstrated on surface advection problems.

Stable and accurate discretizations achieved with hyperviscosity.

Effective on point cloud surface representations.

Abstract

We present a new hyperviscosity formulation for stabilizing radial basis function-finite difference (RBF-FD) discretizations of advection-diffusion-reaction equations on manifolds $\mathbb{M} \subset \mathbb{R}^3$ of co-dimension one. Our technique involves automatic addition of artificial hyperviscosity to damp out spurious modes in the differentiation matrices corresponding to surface gradients, in the process overcoming a technical limitation of a recently-developed Euclidean formulation. Like the Euclidean formulation, the manifold formulation relies on von Neumann stability analysis performed on auxiliary differential operators that mimic the spurious solution growth induced by RBF-FD differentiation matrices. We demonstrate high-order convergence rates on problems involving surface advection and surface advection-diffusion. Finally, we demonstrate the applicability of our formulation to advection-diffusion-reaction equations on manifolds described purely as point clouds. Our surface discretizations use the recently-developed RBF-LOI method, and with the addition of hyperviscosity, are now empirically high-order accurate, stable, and free of stagnation errors.