An Adaptive Projection Method for the Incompressible Navier-Stokes Equations

Ann S. Almgren

John B. Bell

Louis H. Howell

Phillip Colella


In this paper we present a method for solving the time-dependent incompressible Navier-Stokes equations on an adaptive grid. The method is based on a projection formulation in which we first solve convection-diffusion equations to predict intermediate velocities, and then project these velocities onto a space of approximately divergence-free vector fields. Our treatment of convection uses a specialized second-order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for high Reynolds number flows.

Our approach to adaptive refinement uses a nested hierarchy of grids with simultaneous refinement of the grids in both space and time. The integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced, fine grids are advanced multiple steps to reach the same time as the coarse grids, and the grid levels are then synchronized.