A dynamically adaptive multigrid algorithm for the incompressible Navier-Stokes equations—validation and model problems

We describe an algorithm for the solution of the laminar, incompressible Navier-Stokes equations. The basic algorithm is a multigrid method based on a robust, box-based smoothing step. Its most important feature is the incorporation of automatic, dynamic mesh refinement. Using an approximation to the local truncation error to control the refinement, we use a form of domain decomposition to introduce patches of finer grid wherever they are needed to ensure an accurate solution. This refinement strategy is completely local: regions that satisfy our tolerance are unmodified, except when they must be refined to maintain reasonable mesh ratios. This locality has the important consequence that boundary layers and other regions of sharp transition do not “steal” mesh points from surrounding regions of smooth flow, in contrast to moving mesh strategies where such “stealing” is inevitable. Our algorithm supports generalized simple domains, that is, any domain defined by horizontal and vertical lines. This generality is a natural consequence of our domain decomposition approach. We base our program on a standard staggered-grid formulation of the Navier-Stokes equations for robustness and efficiency. To ensure discrete mass conservation, we have introduced special grid transfer operators at grid interfaces in the multigrid algorithm. While these operators complicate the algorithm somewhat, our approach results in exact mass conservation and rapid convergence.