A compact RBF-FD based meshless method for the incompressible Navier–Stokes equations

Description/Abstract

Meshless methods for solving fluid and fluid-structure problems have become a
promising alternative to the finite volume and finite element methods. In this paper, a meshfree
computational method based on radial basis functions in a finite difference mode (RBFFD)
has been developed for the incompressible Navier–Stokes (NS) equations in stream
function vorticity form. This compact RBF-FD formulation generates sparse coefficient
matrices, and hence advancing solutions will in time be of comparatively lower cost. The
spatial discretization of the incompressible NS equations is done using the RBF-FD method
and the temporal discretization is achieved by explicit Euler time-stepping and the Crank–
Nicholson method. A novel ghost node strategy is used to incorporate the no-slip boundary
conditions. The performance of the RBF-FD scheme with the ghost node strategy is validated
against a variety of benchmark problems, including a model fluid–structure interaction
problem, and is found to be in a good agreement with the existing results. In addition, a higherorder
RBF-FD scheme (which uses ideas from Hermite interpolation) is then proposed for
solving the NS equations