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 H¹ 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 H¹ 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.