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We present a novel hyperviscosity formulation for stabilizing RBF-FDdiscretizations of the advection-diffusion equation. The amount ofhyperviscosity is determined quasi-analytically for commonly-used explicit,implicit, and implicit-explicit (IMEX) time integrators by using a simple 1Dsemi-discrete Von Neumann analysis. The analysis is applied to an analyticalmodel of spurious growth in RBF-FD solutions that uses auxiliary differentialoperators mimicking the undesirable properties of RBF-FD differentiationmatrices. The resulting hyperviscosity formulation is a generalization ofexisting ones in the literature, but is free of any tuning parameters and canbe computed efficiently. To further improve robustness, we introduce a simplenew scaling law for polynomial-augmented RBF-FD that relates the degree ofpolyharmonic spline (PHS) RBFs to the degree of the appended polynomial. Whenused in a novel ghost node formulation in conjunction with therecently-developed overlapped RBF-FD method, the resulting method is robust andfree of stagnation errors. We validate the high-order convergence rates of ourmethod on 2D and 3D test cases over a wide range of Peclet numbers (1-1000). Wethen use our method to solve a 3D coupled problem motivated by models ofplatelet aggregation and coagulation, again demonstrating high-orderconvergence rates.

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