Mesh-free multilevel iterative algorithm for Navier–Stokes equations

Abstract

In this article, we developed a computationally efficient multilevel local radial basis function (RBF-FD) mesh-free algorithm. The algorithm provides a new strategy to get good order of accuracy with less computational time, which is most important in the present world. The main idea is the layer-by-layer calculation and then layer-by-layer correction from coarsest level to finest level node points. Numerical experiments are presented to verify the accuracy and efficiency of our developed algorithm with two-dimensional Poisson equation and vorticity–stream function of the incompressible Navier–Stokes equations. The flow inside a lid-driven cavity constitutes a classical benchmark problem, due to its unique boundary conditions that allow comparing any new method’s efficiency for solving Navier–Stokes equations for internal flows. Numerical results are presented through the figures and tables to demonstrate accuracy, efficiency, and convergence of the method. The developed scheme saves at least 60% of CPU time for Poisson equation and 59% of the CPU time for Navier–Stokes equation than the usual local RBF method. The iteration matrix of the proposed local RBF method satisfies the necessary and sufficient condition for convergence.