Stabilized Meshless Local Petrov-Galerkin (MLPG) method for incompressible viscous fluid flows

In this paper, the truly Meshless Local Petrov-Galerkin (MLPG) method is extended for computation of steady incompressible flows, governed by the Navier-Stokes equations (NSE), in vorticity-stream function formulation. The present method is a truly meshless method based on only a number of randomly located nodes. The formulation is based on two equations including stream function Poisson equation and vorticity advection-dispersion-reaction equation (ADRE). The meshless method is based on a local weighted residual method with the Heaviside step function and quartic spline as the test functions respectively over a local subdomain. Radial basis functions (RBF) interpolation is employed in shape function and its derivatives construction for evaluating the local weak form integrals. Due to satisfaction of kronecker delta property in RBF interpolation, no special technique is employed to enforce the essential boundary conditions. In order to overcome instability and numerical errors (numerical dispersion) encountering in convection dominant flows, a new up-winding scheme is introduced and used to stabilize the convection operator in the streamline direction (as is done in SUPG). In this upwinding technique, instead of moving subdomains, the weight function is shifted in the direction of flow. Efficiency and accuracy of the new stabilization technique are investigated by a problem and compared with other stabilization techniques. In order to obtain the optimum parameters, Shape parameters of Multiquadric-RBF for both poisson and convection-diffusion equations are tuned and studied. Effects of subdomain (integration domain) and support domain sizes are also studied. The efficiency, accuracy and robustness of the modified MLPG are demonstrated by a well-known benchmark test problem including the standard lid driven cavity flow.