A Gaussian–cubic backward substitution method for the four-order pure stream function formulation of two-dimensional incompressible viscous flows

In this study, a novel meshless collocation method based on the Gaussian–cubic hybrid kernel function in conjunction with the ghost-points method and the general Newton–Raphson method is proposed for solving the four-order stream function formulation of Navier–Stokes equations. The decrease of variables and equations results in better computational efficiency of stream function formulation than primitive variable formulations of 2D incompressible viscous flow. The original Naiver–Stokes equations can be transformed into a four-order stream function partial differential equation by introducing the vorticity and stream function. The new proposed Gaussian–cubic backward substitution method is used to solve the corresponding system of equations where the nonlinear stream function formulation is linearized by the general Newton–Raphson method. The ghost-points method is applied as the layout scheme of center points, which can effectively improve the accuracy without requiring more computational resources. Several examples are provided and the results demonstrate the feasibility of the proposed novel approach.