Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection–diffusion equations

Abstract

We present a novel hyperviscosity formulation for stabilizing RBF-FD discretizations of the advectiondiffusion 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.

Node Generation for RBF-FD Methods by QR Factorization

“…Hyperviscosity methods were first introduced in [39] and further studied in [40,41]. These methods allow stable numerical time-stepping for RBF-FD methods.…”

Robust Node Generation for Mesh-free Discretizations on Irregular Domains and Surfaces

“…Biological problems often involve domains with not only irregular outer boundaries, but irregular embedded inner boundaries, e.g., platelets and red blood cells in a blood vessel [34]. While the authors have begun developing numerical methods for simulations in such complex domains [36,31,32], our methods currently lack a robust node generation algorithm that can modify its node sets locally when the embedded inner boundaries deform, and are added or removed. Fortunately, Algorithm 1 is easily modified to tackle this problem.…”

“…Boundary refinement and ghost nodes. The numerical solution of PDEs with RBF-FD sometimes requires denser node sets near domain boundaries [11,12,31], and/or ghost nodes outside the domain boundary to enforce boundary conditions [11,1,32]. Our node generator must therefore possess these capabilities.…”

“…For the spatial discretization, we use the overlapped RBF-FD method [31], and the ghost node technique outlined in [32]. The RBF-FD discretization is achieved by using PHS RBFs in conjunction with high-degree polynomials, where the convergence rates are dictated purely by the polynomial degree [12,11,1].…”

The boundary knot method for the solution of two-dimensional advection reaction-diffusion and Brusselator equations

“…Besides, as described in Murray (1993), because of Tyson’s observation, the steady state patterns exist in at least part of the indeterminate region, where the analysis cannot predict what patterns will occur. In recent years, because of the lack of analytical solutions for evaluating chemotaxis models such as equation (1), different numerical methods were developed, which contain discontinuous Galerkin method (Epshteyn and Kurganov, 2008), various finite difference (FD) schemes and finite volume methods (Chertock and Kurganov, 2008; Chiu and Yu, 2007; Smiely, 2009; Tyson et al , 1999; Tyson et al , 2000), semi-analytical methods for simulation pattern formation in liquid drops (Dehghan et al , 2012), variational iteration method for solving reaction–diffusion equation (Wu et al , 2015), a finite element method to approximate systems of nonlinear reaction–diffusion–advection equations (Sheshachala and Codina, 2018), local radial basis functions (RBFs) for the solution of parabolic–parabolic Patlak–Keller–Segel chemotaxis model (Dehghan and Abbaszadeh, 2015), RBF-generated finite difference (RBF-FD) scheme for solving reaction–diffusion equations on the surfaces (Shankar et al , 2014; Shankar et al , 2015) and using hyperviscosity-based stabilization in RBF-FD to find the numerical solution of advection–diffusion equations (Shankar and Fogelson, 2018).…”