  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.
