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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:372

Article

Shankar, Varun^1,2  Fogelson, Aaron L.^1,3 

[1] Univ Utah, Dept Math, Salt Lake City, UT 84112 USA

[2] Univ Utah, Sch Comp, Salt Lake City, UT 84112 USA

[3] Univ Utah, Dept Bioengn, Salt Lake City, UT 84112 USA

关键词: Radial basis function;    High-order method;    Hyperviscosity;    Meshfree;    Advection-diffusion;   

DOI  :  10.1016/j.jcp.2018.06.036

来源: Elsevier

【 摘 要 】

We present a novel hyperviscosity formulation for stabilizing RBF-FD discretizations of the advection-diffusion equation. The amount of hyperviscosity is determined quasianalytically 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. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free