High order wide and compact schemes for the steady incompressible Navier-Stokes equations

This thesis is concerned with solving the steady two dimensional Navier-Stokes equations using finite difference methods. It has been discovered that although central difference approximations are locally second-order accurate they often suffer from computational instability and the resulting solutions exhibit nonphysical oscillations. Although first-order and second-order upwind difference approximations are computationally stable, the resulting solutions exhibit the effects of artificial viscosity. As a result of this, there has been great interest in recent years to investigate high-order schemes. In this thesis, two fourth-order accurate finite difference schemes are obtained for the Navier-Stokes equations expressed in streamfunction alone. The first one is the conventional fourth-order central difference scheme with a stencil extending over 29 points. The second one is of compact type with a stencil extending over a 5 x 5-square of points. This method is more efficient for solving the discrete non-linear system by Newton's method. We consider a number of test problems, including the driven cavity problem, to compare the two fourthorder schemes to each other, as well as second-order and fourth-order benchmark solutions. In spite of its wider stencil, it is found that the conventional scheme has sufficiently lower error for high Reynolds number than the compact scheme, making up for its greater width. The effects of numerical boundary conditions on convergence are also investigated.