We present a new hyperviscosity formulation for stabilizing radial basis function-finite difference (RBF-FD) discretizations of advection-diffusion-reaction equations on manifolds 𝕄 ⊂ ℝ3 of codimension 1. 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-least orthogonal interpolation method and, with the addition of hyperviscosity, are now empirically high-order accurate, stable, and free of stagnation errors.
This document was originally published in SIAM Journal on Scientific Computing by Society for Industrial and Applied Mathematics (SIAM). Copyright restrictions may apply. doi: 10.1137/19M1288747
Shankar, Varun; Wright, Grady B.; and Narayan, Akil. (2020). "A Robust Hyperviscosity Formulation for Stable RBF-FD Discretizations of Advection-Diffusion-Reaction Equations on Manifolds". SIAM Journal on Scientific Computing, 42(4), A2371-A2401. https://dx.doi.org/10.1137/19M1288747