Bivariate Spline Method for Numerical Solution of Time Evolution Navier-Stokes Equations over Polygons in Stream Function Formulation

We use the bivariate spline method to solve the time evolution Navier-Stokes equations numerically. The bivariate spline we use in this paper is the spline space of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the Navier-Stokes equations is employed. Galerkin's method is applied to discretize the space variables of the nonlinear fourth order equation, Crank-Nicholson's method is applied to discretize the time variable, and Newton's iterative method is then used to solve the resulting nonlinear system. We show the existence and uniqueness of the weak solution in L 2 (0; T; H 2 (()) \ L 1 (0; T; H 1 (()) of the 2D nonlinear fourth order problem and give an estimate of how fast the numerical solution converges to the weak solution. The C 1 cubic splines are implemented in MATLAB for solving the Navier-Stokes equations numerically. Our numerical experiments show that the method is eeective and eecient. x1. Introduction We are interested in using bivariate spline functions to numerically solve the time evolution Navier-Stokes' equations over a planar polygon. The aim of this research is to provide an eecient numerical tool for uid dynamics simulation. This is a continuation of our eeort Lai and Wenston'97] where we considered the bivariate spline method for numerical solution of the steady-state Navier-Stokes equations. Let R 2 be a simply connected polygonal domain and u = (u 1 ; u 2) T be the planar velocity of a uid ow over. Also, let p be the pressure function, f = (f 1 ; f 2) T be the external body force of the uid, h = (h 1 ; h 2) T the velocity of the uid ow on the boundary @ and g = (g 1 ; g 2) T an initial velocity. Then the time evolution Navier-Stokes equations are