An adaptive multilevel multigrid formulation for Cartesian hierarchical grid methods

Abstract

A Cartesian grid method with adaptive mesh refinement and multigrid acceleration is presented for the compressible Navier–Stokes equations. Cut cells are used to represent boundaries on the Cartesian grid, while ghost cells are introduced to facilitate the implementation of boundary conditions. A cell-tree data structure is used to organize the grid cells in a hierarchical manner. Cells of all refinement levels are present in this data structure such that grid level changes as they are required in a multigrid context do not have to be carried out explicitly. Adaptive mesh refinement is introduced using phenomenon-based sensors. The application of the multilevel method in conjunction with the Cartesian cut-cell method to problems with curved boundaries is described in detail. A 5-step Runge–Kutta multigrid scheme with local time stepping is used for steady problems and also for the inner integration within a dual time-stepping method for unsteady problems. The inefficiency of customary multigrid methods on Cartesian grids with embedded boundaries requires a new multilevel concept for this application, which is introduced in this paper. This new concept is based on the following novelties: a formulation of a multigrid method for Cartesian hierarchical grid methods, the concept of averaged control volumes, and a mesh adaptation strategy allowing to directly control the number of refined and coarsened cells.