GCR(k)-multigrid solver for the anelastic equations of atmospheric flow in general curvilinear coordinates

Steve Thomas

and

Piotr Smolarkiewicz

National Center for Atmospheric Research
Mesoscale and Microscale Meteorology (MMM) Division
Foothills Laboratory, Boulder Colorado


Abstract

A nonoscillatory forward-in-time (NFT) discretisation of the anelastic Euler/Navier Stokes equations in general curvilinear coordinates results in an elliptic pressure equation with a non self-adjoint operator (Smolarkiewicz and Margolin 1997). To facilitate the implementation of both flux-form Eulerian and monotone semi-Lagrangian advection schemes, an unstaggered Arakawa A-grid is employed and thus line relaxation schemes must be based on pentadiagonal solvers. We describe a flexible Generalized Conjugate Residual GCR(k) Krylov type solver (Smolarkiewicz and Margolin 1994) which allows for inexact or multi-level preconditioning. A reduced operator is combined with a simple multigrid V-cycle as the preconditioner. We employ line Jacobi, SOR and ADI type smoothers, similar to the incomplete ILU approach presented at the 7th Copper Mountain Conference by Vuik, Wesseling and Zeng. We present numerical simulations illustrating the effect of steep terrain on the solver convergence rate.

[1] P. K. Smolarkiewicz and L. G. Margolin. On forward-in-time differencing for fluids: An Eulerian/semi-Lagrangian nonhydrostatic model for stratified flows. Atmos. Ocean, in press (1997).

[2] P. K. Smolarkiewicz and L. G. Margolin. Variational solver for elliptic problems in atmospheric flows. Appl. Math. and Comp. Sci., vol 4 (1994), pp. 527-551.

[3] C. Vuik, P. Wesseling, S. Zeng. Krylov subspace and multigrid methods applied to the incompressible Navier-Stokes equations. Proceedings of the 7th Copper Mountain Conference on Multigrid Methods, 1995, pp. 737-753.