Stable nodal projection method on octree grids - Part I: Analytical Results

We introduce a novel projection method to solve the incompressible Navier-Stokes equations with arbitrarily shaped boundaries. Our method employs an adaptive mesh refinement strategy using non-graded quad/octree grids. The viscosity and projection steps are discretized using supra-convergent finite difference approximations with sharp boundary treatment. The novelty of our method comes from the fact that we collocate the velocity and pressure terms in the projection step. By collocating the projection step, we reduce the complexity in the underlying data structures used and streamlines the code development. In Part I of this two part presentation, we focus on the stability analysis of our method through a combination of proofs and numerical studies. We show that our projection operator yields second order accuracy and is stable in the presence of a variety of boundary and interface conditions.