Preconditioned conjugate-residual solvers for Helmholtz equations in nonhydrostatic models

Numerical integration of the compressible nonhydrostatic equations using semi-implicit techniques is complicated by the need to solve a Helmholtz equation at each time step. The authors present an accurate and efficient technique for solving the Helmholtz equation using a conjugate-residual (CR) algorithm that is accelerated by ADI preconditioned. These preconditioned CR solvers possess four distinct advantages over most other solvers that have been used with the Helmholtz equations that arise in compressible nonhydrostatic semi-implicit atmospheric models: the preconditioned CR methods 1) can solve Helmholtz equations containing variable coefficients, alleviating the need to prescribe a reference state in order to simplify the elliptic problem; 2) transparently include the cross-derivative terms arising from terrain transformations; 3) are efficient and accurate for nonhydrostatic models used across a broad range of scales, from cloud scales to synoptic-global scales; and 4) are easy to formulate and program. These features of the CR solver allow semi-implicit formulations that are unconstrained by the form of the Helmholtz equations, and the authors propose a formulation that is more consistent than those most often used in that it includes implicit treatment of all terms associated with the pressure gradients and divergence. This formulation is stable for nonhydrostatic-scale simulations involving steep terrain, whereas the more common semi-implicit formulation is not. The ADI preconditioners are presented for use in simulations of both hydrostatic and nonhydrostatic scale flows. These simulations demonstrate the efficiency and accuracy of the preconditioned 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.