Numerical solutions of two-dimensional flow fields by using the localized method of approximate particular solutions

A combination of the localized method of approximate particular solutions (LMAPS), the implicit Euler method and the Newton’s method is adopted in this paper for transient solutions of two-dimensional velocity–vorticity formulation of the Navier–Stokes equations. The LMAPS, which is truly free from time-consuming mesh generation and numerical quadrature, and the implicit Euler method are, respectively, used for spatial and temporal discretizations of the velocity–vorticity formulation. Using the approximations of particular solutions in every local domain, the derivatives at nodes with respect to space coordinates via the LMAPS can be approximated by linear summations of nearby function values. After the discretizations for space and time derivatives, a system of nonlinear algebraic equations will be yielded at every time step and then the Newton’s method is used for efficiently analyzing these systems. Three numerical examples are provided to validate the accuracy and the simplicity of the proposed scheme and the numerical results are compared well with other numerical and analytical solutions. Besides, the numerical solutions, acquired by using different numbers of total nodes, different numbers of nodes in sub-domain, different shape parameters and different Reynolds numbers, are provided to show the merits of the proposed meshless scheme.