An RBF-based finite difference discretization of the Navier-Stokes equations: error analysis and applications

Radial basis function-finite differences (RBF-FD) are used to solve the incompressible Navier-Stokes equations on scattered nodes. We present a semi-implicit fractional-step method that uses a staggered grid arrangement. The RBF-QR method devised by Fornberg&Piret (2008) is used to obtain the RBF-FD weights for the spatial derivatives. We propose a rigorous error analysis strategy to identify optimal combinations of the shape parameter, Ɛ, and the stencil size, n. A modified wavenumber analysis shows that the accuracy of the RBF differentiation matrices based on the optimal parameters is comparable to 4th-order Padé-type finite differences for both first and second derivatives. The internal flow in a lid-driven cavity and a transient cylinder wake are studied as examples. The former uses quasi-uniform scattered nodes, while the latter refines the mesh near the cylinder and on the wake centerline. We demonstrate that stable solutions are obtained without the need for hyperviscosity.