Convergence of a Compact Scheme for the Pure Streamfunction Formulation of the Unsteady Navier–Stokes System

Abstract

This paper is devoted to the analysis of a new compact scheme for the Navier–Stokes equations in pure streamfunction formulation. Numerical results using that scheme have been reported in [M. Ben‐Artzi et al., J. Comput. Phys., 205 (2005), pp. 640–664]. The scheme discussed here combines the Stephenson scheme for the biharmonic operator and ideas from box‐scheme methodology. Consistency and convergence are proved for the full nonlinear system. Instead of customary periodic conditions, the case of boundary conditions is addressed. It is shown that in one dimension the truncation error for the biharmonic operator is $O(h^4)$ at interior points and $O(h)$ at near‐boundary points. In two dimensions the truncation error is $O(h^2)$ at interior points (due to the cross‐terms) and $O(h)$ at near‐ boundary points. Hence the scheme is globally of order four in the one‐dimensional periodic case and of order two in the two‐dimensional periodic case, but of order 3/2 for one‐ and two‐ dimensional nonperiodic boundary conditions. We emphasize in particular that there is no special treatment of the boundary, thus allowing robust use of the scheme. The finite element analogy of the finite difference schemes is invoked at several stages of the proofs in order to simplify their verifications.