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

arXiv: https://arxiv.org/abs/1910.07059 · HTML: https://mlsanity.com/paper/1910.07059/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Oct 2019.

## Built on by (most cited first, 1 of 1)

- [Accelerated Training of Physics-Informed Neural Networks (PINNs) using Meshless Discretizations](https://mlsanity.com/paper/2205.09332/) (2022)

## Abstract (arXiv)

We present a new hyperviscosity formulation for stabilizing radial basis function-finite difference (RBF-FD) discretizations of advection-diffusion-reaction equations on manifolds mathbbM subset ℝ3 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.