A meshless generalized finite difference scheme for the stream function formulation of the Naiver-Stokes equations

Abstract

In this study, a meshless numerical scheme, which is the combination of the generalized finite difference method, the fictitious-nodes technique, and the two-step Newton-Raphson method, was proposed to solve the stream function formulation of the Navier-Stokes equations. To reduce the numbers of unknowns and partial differential equations for the two-dimensional flow field of a viscous incompressible fluid, the stream function formulation, which is a fourth-order nonlinear partial differential equation with only one unknown variable, can be derived by introducing the definition of vorticity and stream function. By applying the generalized finite difference method, the derivatives in the stream function formulation can be simply expressed as a linear combination of functional data and weighting values at several nearest nodes. To deal with the two boundary conditions for the fourth-order partial differential equations, the fictitious-nodes technique was utilized to set up the fictitious nodes outside the computational domain, and cooperate with the generalized finite difference method to form a system of nonlinear algebraic equations. The aforementioned numerical procedure can yield a square Jacobian matrix, so the Newton-Raphson method can be adopted to efficiently acquire the numerical solutions. Several numerical examples were tested to verify the practicability of the proposed meshless numerical scheme.