Numerical simulation of time-dependent Navier–Stokes and MHD equations using a meshless method based on fundamental and particular solutions

Abstract

In this paper a meshless method based on fundamental and particular solution (MFS–MPS) is implemented to numerically solve the time-dependent Navier–Stokes equations in stream function–vorticity form for lid-driven cavity flows. Further, the method is applied to natural convection problem in a cavity where an additional temperature equation and mixed boundary conditions are involved. Finally the MHD equations in stream function–vorticity–magnetic field-current density form are solved for MHD flows in a lid-driven cavity. A semi-implicit approach is used for the time advancing in which the time derivative is discretized using first order forward-difference approximation, the Laplace operator is taken in next time level, and rest of the terms are taken in the current time level. We take the number of boundary collocation points more than the source points and solve the overdetermined system of equation in a least squares sense at each time step. The least squares approach alleviates the problem of ill-conditioning to a certain extent. The results obtained are in good agreement with the previous numerical works where available. We find that the meshless method based on MFS–MPS is simple and effective, and can easily be applied to the coupled time-dependent nonlinear system of equations.