Solving the incompressible fluid flows by a high-order mesh-free approach

In this paper, we propose for the first time to extend the application field of the high-order mesh-free approach to the stationary incompressible Navier-Stokes equations. This approach is based on a high-order algorithm, which combines a Taylor series expansion, a continuation technique, and a moving least squares (MLS) method. The Taylor series expansion permits to transform the nonlinear problem into a succession of continuous linear ones with the same tangent operator. The MLS method is used to transform the succession of continuous linear problems into discrete ones. The continuation technique allows to compute step-by-step the whole solution of the discrete problems. This mesh-free approach is tested on three examples: a flow around a cylindrical obstacle, a flow in a sudden expansion, and the standard benchmark lid-driven cavity flow. A comparison of the obtained results with those computed by the Newton-Raphson method with MLS, the high-order continuation with finite element method, and those of literature is presented.