ADI+MDA orthogonal spline collocation for the pressure Poisson reformulation of the Navier-Stokes equation in two space variables

Nick Fisher

Abstract: A numerical method for solving a pressure Poisson reformulation of the Navier–Stokes equation in two space variables is presented. The method discretizes in space using orthogonal spline collocation with splines of order r<math><mi is="true">r</mi></math>. The velocity terms are obtained through an alternating direction implicit extrapolated Crank –Nicolson scheme applied to a Burgers’ type equation and the pressure term is found by applying a matrix decomposition algorithm to a Poisson equation satisfying non-homogeneous Neumann boundary conditions at each time level. Numerical results suggest that the scheme exhibits convergence rates of order r<math><mi is="true">r</mi></math> in space in the H1<math><msup is="true"><mrow is="true"><mi is="true">H</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msup></math> norm and semi-norm for the velocity and pressure terms, respectively, and is order 2 in time. Finally, the scheme is applied to the lid-driven cavity problem and is compared to standard benchmark values.