A mass-conservative adaptive FAS multigrid solver for cell-centered finite difference methods on block-structured, locally-cartesian grids

•We present a new conservative FAS multigrid solver for finite difference methods on AMR grids.•The method can be applied in any number of dimensions and can be easily modified for nonlinear, time-dependent, and coupled systems of equations.•The smoother, prolongation, and restriction operations need never be aware of the mass conservation conditions.•We demonstrate that the solver has optimal, or nearly optimal, complexity.

We present a mass-conservative full approximation storage (FAS) multigrid solver for cell-centered finite difference methods on block-structured, locally cartesian grids. The algorithm is essentially a standard adaptive FAS (AFAS) scheme, but with a simple modification that comes in the form of a mass-conservative correction to the coarse-level force. This correction is facilitated by the creation of a zombie variable, analogous to a ghost variable, but defined on the coarse grid and lying under the fine grid refinement patch. We show that a number of different types of fine-level ghost cell interpolation strategies could be used in our framework, including low-order linear interpolation. In our approach, the smoother, prolongation, and restriction operations need never be aware of the mass conservation conditions at the coarse-fine interface. To maintain global mass conservation, we need only modify the usual FAS algorithm by correcting the coarse-level force function at points adjacent to the coarse-fine interface. We demonstrate through simulations that the solver converges geometrically, at a rate that is h-independent, and we show the generality of the solver, applying it to several nonlinear, time-dependent, and multi-dimensional problems. In several tests, we show that second-order asymptotic (h→0) convergence is observed for the discretizations, provided that (1) at least linear interpolation of the ghost variables is employed, and (2) the mass conservation corrections are applied to the coarse-level force term.