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

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. 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 in space in the H1 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.