Orthogonal spline collocation methods for fluid flow problems

Abstract

We propose an approach for the numerical solution of the Navier-Stokes equations based on a pressure Poisson equation reformulation. Through an alternating direction implicit extrapolated Crank--Nicolson time discretization, the scheme decouples the updates for velocity and pressure terms. Moreover, the proposed scheme reduces the Navier-Stokes equations to a Burgers' equation for the velocity terms and a singular Neumann Poisson equation for the pressure. These two sub-problems are analyzed in turn. We use extrapolated alternating direction implicit Crank-Nicolson orthogonal spline collocation with splines of order $r$ to solve the coupled Burgers' equations in two space variabl and two unknown functions. The scheme is initialized with an alternating direction implicit predictor-corrector method. We show theoretically that the $H^1$ norm of the error at each time level is of order $r$ in space and of order 2 in time. Then we use a matrix decomposition algorithm for the orthogonal spline collocation solution to Poisson's equation with Neumann boundary conditions. We show theoretically that the $H^1$ semi-norm of the error is of order $r$. In each case, our numerical results confirm these theoretical orders. Finally, the combined scheme is implemented for the solution of the pressure Poisson reformulation of the Navier--Stokes equations using splines of equal order. Numerical results show that the scheme obtains the expected optimal order convergence rates for both the velocity and pressure terms.