Hyperviscosity-Based Stabilization for Radial Basis Function-Finite Difference (RBF-FD) Discretizations of Advection-Diffusion Equations

TL;DR

This paper introduces a parameter-free hyperviscosity stabilization technique for RBF-FD discretizations of advection-diffusion equations, validated through high-order convergence in 2D and 3D tests, including complex biological models.

Contribution

It develops a quasi-analytical hyperviscosity formulation for RBF-FD that is parameter-free, efficient, and enhances robustness when combined with a ghost node approach and overlapped RBF-FD.

Findings

Achieves high-order convergence in 2D and 3D tests

Demonstrates robustness across a wide range of Peclet numbers

Successfully applies to complex biological models of platelet aggregation

Abstract

We present a novel hyperviscosity formulation for stabilizing RBF-FD discretizations of the advection-diffusion equation. The amount of hyperviscosity is determined quasi-analytically for commonly-used explicit, implicit, and implicit-explicit (IMEX) time integrators by using a simple 1D semi-discrete Von Neumann analysis. The analysis is applied to an analytical model of spurious growth in RBF-FD solutions that uses auxiliary differential operators mimicking the undesirable properties of RBF-FD differentiation matrices. The resulting hyperviscosity formulation is a generalization of existing ones in the literature, but is free of any tuning parameters and can be computed efficiently. To further improve robustness, we introduce a simple new scaling law for polynomial-augmented RBF-FD that relates the degree of polyharmonic spline (PHS) RBFs to the degree of the appended polynomial. When used in a novel ghost node formulation in conjunction with the recently-developed overlapped RBF-FD method, the resulting method is robust and free of stagnation errors. We validate the high-order convergence rates of our method on 2D and 3D test cases over a wide range of Peclet numbers (1-1000). We then use our method to solve a 3D coupled problem motivated by models of platelet aggregation and coagulation, again demonstrating high-order convergence rates.