A FIRST STEP IN APPLYING THE SINC COLLOCATION METHOD TO THE NONLINEAR NAVIER-STOKES EQUATIONS

Abstract

Different numerical approaches have been proposed in the past to solve the Navier-Stokes equations. Conventional methods have often relied on finite-difference, finite-element, and boundary-element techniques. Multigrid methods have been recently introduced because they help to obtain a faster convergence rate of the error residual. A difficulty plaguing numerical methods today is the inability to treat singularities at or near boundaries. Such difficulties become even more pronounced when coupled with the need to handle semi-infinite and infinite domains. Sinc-based numerical algorithms have the advantage of handling singularities, boundary layers, and semi-infinite domains very effectively. In addition, they typically require fewer nodal points and are proven to provide an exponential convergence rate in solving linear differential equations. This study involves a first step in applying the Sinc-based algorithm to solve a nonlinear set of partial differential equations. The example we consider arises in the context of a driven-cavity flow in two space dimensions. As such, the steady and incompressible Navier-Stokes equations are solved by means of two-dimensional Sinc collocation in conjunction with the primitive variable method and a pressure correction algorithm based on artificial compressibility. Simulations are also carried out using forward differences, central differences, and a commercial code. Results are compared with one another and with the Sinc-collocation approximation. It is found that the error in the Sinc-collocation approximation outperforms other solutions, especially near the singular corners of the cavity.