A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations

TL;DR

This paper introduces a novel RBF-based projection method for solving incompressible unsteady Stokes equations in irregular geometries, achieving high-order accuracy without traditional pressure boundary conditions.

Contribution

It develops a new divergence-free RBF technique for the Leray-Helmholtz projection, enabling boundary component matching and eliminating the need for time-splitting or pressure boundary conditions.

Findings

Achieves 5th to 6th order spatial convergence.

Up to 4th order temporal accuracy.

Effective in irregular 2D geometries.

Abstract

A new projection method based on radial basis functions (RBFs) is presented for discretizing the incompressible unsteady Stokes equations in irregular geometries. The novelty of the method comes from the application of a new technique for computing the Leray-Helmholtz projection of a vector field using generalized interpolation with divergence-free and curl-free RBFs. Unlike traditional projection methods, this new method enables matching both tangential and normal components of divergence-free vector fields on the domain boundary. This allows incompressibility of the velocity field to be enforced without any time-splitting or pressure boundary conditions. Spatial derivatives are approximated using collocation with global RBFs so that the method only requires samples of the field at (possibly scattered) nodes over the domain. Numerical results are presented demonstrating high-order convergence in both space (between 5th and 6th order) and time (up to 4th order) for some model problems in two dimensional irregular geometries.