A new fourth-order compact scheme for the Navier–Stokes equations in irregular domains

Abstract

We present a high-order finite difference scheme for Navier–Stokes equations in irregular domains. The scheme is an extension of a fourth-order scheme for Navier–Stokes equations in streamfunction formulation on a rectangular domain (Ben-Artzi et al., 2010). The discretization offered here contains two types of interior points. The first is regular interior points, where all eight neighboring points of a grid point are inside the domain and not too close to the boundary. The second is interior points where at least one of the closest eight neighbors is outside the computational domain or too close to the boundary. In the second case we design discrete operators which approximate spatial derivatives of the streamfunction on irregular meshes, using discretizations of pure derivatives in the x, y and along the diagonals of the element.