RBF-FD discretization of the Navier-Stokes equations on scattered but staggered nodes

TL;DR

This paper introduces a semi-implicit RBF-FD method with a staggered node layout for solving incompressible Navier-Stokes equations, achieving high accuracy and stability on scattered nodes without special wall treatments.

Contribution

It develops a novel RBF-FD discretization with optimized parameters and demonstrates its effectiveness on benchmark fluid flow problems at various Reynolds numbers.

Findings

Achieves spectral-like accuracy with 28-point stencils.

Enables stable, accurate simulations at lower resolutions.

No need for hyperviscosity or special wall treatments.

Abstract

A semi-implicit fractional-step method that uses a staggered node layout and radial basis function-finite differences (RBF-FD) to solve the incompressible Navier-Stokes equations is developed. Polyharmonic splines (PHS) with polynomial augmentation (PHS+poly) are used to construct the global differentiation matrices. A systematic parameter study identifies a combination of stencil size, PHS exponent, and polynomial degree that minimizes the truncation error for a wave-like test function on scattered nodes. Classical modified wavenumber analysis is extended to RBF-FDs on heterogeneous node distributions and used to confirm that the accuracy of the selected 28-point stencil is comparable to that of spectral-like, 6th-order Pad\'e-type finite differences. The Navier-Stokes solver is demonstrated on two benchmark problems, internal flow in a lid-driven cavity in the Reynolds number regime $10^2\leq$Re$\leq10^4$, and open flow around a cylinder at Re=100 and 200. The combination of grid staggering and careful parameter selection facilitates accurate and stable simulations at significantly lower resolutions than previously reported, using more compact RBF-FD stencils, without special treatment near solid walls, and without the need for hyperviscosity or other means of regularization.