Wc. Skamarock et al., PRECONDITIONED CONJUGATE-RESIDUAL SOLVERS FOR HELMHOLTZ EQUATIONS IN NONHYDROSTATIC MODELS, Monthly weather review, 125(4), 1997, pp. 587-599
Numerical integration of the compressible nonhydrostatic equations usi
ng semi-implicit techniques is Complicated by the need to solve a Helm
holtz equation at each time step. The authors present an accurate and
efficient technique for solving the Helmholtz equation using a conjuga
te-residual (CR) algorithm that is accelerated by-ADI preconditioners.
These preconditioned CR solvers possess four distinct advantages over
most other solvers that have been used with the Helmholtz equations t
hat arise in compressible nonhydrostatic semi-implicit atmospheric mod
els: the preconditioned CR methods 1) can solve Helmholtz equations co
ntaining variable coefficients, alleviating the need to prescribe a re
ference state in order to simplify the elliptic problem; 2) transparen
tly include the cross-derivative terms arising from terrain transforma
tions; 3) are efficient and accurate for nonhydrostatic models used ac
ross a broad range of scales, from cloud scales to synoptic-global sca
les; and 4) are easy to formulate and program. These features of the C
R solver allow semi-implicit formulations that are unconstrained by th
e form of the Helmholtz equations, and the authors propose a formulati
on that is more consistent than those most often used in that it inclu
des implicit treatment of all terms associated with the pressure gradi
ents and divergence. This formulation is stable for nonhydrostatic-sca
le simulations involving steep terrain, whereas the more common semi-i
mplicit formulation is not. The ADI preconditioners are presented for
use in simulations of both hydrostatic and nonhydrostatic scale hows.
These simulations demonstrate the efficiency and accuracy of the preco
nditioned CR method and the overall stability of the model formulation
. The simulations also suggest a general convergence criteria for the
iterative algorithm in terms of the solution divergence.