PRECONDITIONED CONJUGATE-RESIDUAL SOLVERS FOR HELMHOLTZ EQUATIONS IN NONHYDROSTATIC MODELS

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.