Combination of Meshless Local Petrov-Galerkin and Finite Difference Methods for Analysis of Transient and Incompressible Navier–Stokes Equations

This paper presents a numerical algorithm for solving unsteady viscous incompressible two–dimensional (2D) Navier–Stokes equations. In the proposed method, for discretization of time derivatives and solving the Poisson equation of the pressure, Meshless Local Petrov-Galerkin (MLPG) and forward finite difference methods are employed, respectively. In the present analysis, the moving least-square (MLS) approximation is regarded for interpolation, and the Gaussian weight function is used as the test function. To satisfy the boundary conditions, the penalty approach is applied. In the numerical examples, the accuracy and efficiency of the method are compared with those of the exact solutions. The effects of the number of nodes, the size of time interval, as well as the nodes distribution (both regular and irregular) on the relative errors are investigated. Moreover, the Gaussian integral sub-domains with the circular and square shapes are considered, and the accuracy of the results is compared with each other. Analysis of these results for 2D benchmark geometries with different boundary conditions clearly displays that the accuracy of the suggested combined method for solution of the problem related to unsteady viscous incompressible 2D flows is high such that its differences with analytical solution is negligible. Since no limitations is considered on the design process of the regarded numerical algorithm; therefore, it is respected that this approach is successful and has sufficient efficiency to solve the governing equations.