The reproducing kernel particle Petrov–Galerkin method for solving two-dimensional nonstationary incompressible Boussinesq equations

Abstract

The meshless techniques are improved to simulate a wide range of the physical models. Two common forms of the meshless methods are strong and (local) weak forms. In the current paper, we employ the local weak form technique to simulate the two-dimensional non-stationary Boussinesq equations. In the present method the trial functions have been selected from the shape functions of the RKPM. Also, the test function is based upon a Ck function. At first, the time variable has been discretized via a finite difference scheme and then the space direction has been approximated by using the MLPG procedure. In this procedure, by employing the continuity equation, the two-dimensional non-stationary Boussinesq equations have been transformed to a pressure Poisson equation. After solving the obtained pressure Poisson equation, the velocity of fluid in the x- and y-directions and also the temperature can be updated, directly. Numerical examples confirm the ability of the developed technique.