Numerical study of orthogonal spline collocation methods for unsteady incompressible Navier-Stokes equations

Abstract

In this thesis, I shall present a numerical investigation of unsteady incompressible Navier-Stokes equations by using a so-called orthogonal spline collocation method. The method is based on the use of Hermite bi-cubic spline approximation in spatial directions and a two-step leapfrog type approximation in the time direction. When the solution is smooth, it has been proved theoretically that the method has the accuracy O(r2 + h3) in H¹-norm where T and h are the stepsizes at time direction and spatial directions, respectively. The numerical results in this thesis confirm the previous theoretical analysis with both uniform and non-uniform meshes. The numerical experiments also demonstrate the as yet unproved phenomenon of superconvergence, which frequently occurs in orthogonal spline collocation methods but so far the only proof is for Poisson's equation in the unit square. Finally, the method is applied for solving the driven cavity problem with both smooth and non-smooth started flow, which has been studied by many people with finite element methods, finite difference methods, compact finite difference schemes and spectral methods. The efficient implementation and sparse structure of method are studied. The numerical experiments show that the spline collocation method is efficient and less cost, comparing with those classical methods.