A second order accurate projection method for the incompressible Navier–Stokes equations on non-graded adaptive grids

Abstract

We present an unconditionally stable second order accurate projection method for the incompressible Navier–Stokes equations on non-graded adaptive Cartesian grids. We employ quadtree and octree data structures as an efficient means to represent the grid. We use the supra-convergent Poisson solver of [C.-H. Min, F. Gibou, H. Ceniceros, A supra-convergent finite difference scheme for the variable coefficient Poisson equation on fully adaptive grids, CAM report 05-29, J. Comput. Phys. (in press)], a second order accurate semi-Lagrangian method to update the momentum equation, an unconditionally stable backward difference scheme to treat the diffusion term and a new method that guarantees the stability of the projection step on highly non-graded grids. We sample all the variables at the grid nodes, producing a scheme that is straightforward to implement. We propose two and three-dimensional examples to demonstrate second order accuracy for the velocity field and the divergence free condition in the L1 and L∞ norms.