Parallel Preconditioned GMRES Solvers for 3-D Helmholtz Equations in Regional Non-hydrostatic Atmosphere Model

The semi-implicit time stepping scheme in non-hydrostatic compressible atmosphere model makes it necessary to solve 3-D helmholtz equations, which are complicated with variable coefficients and cross derivative terms. Since the ill-conditioned matrix is nonsymmetric, preconditioned GMRES Krylov iterative algorithm is adopted. Based on PETSc and Hypre parallel package, we make comparison between several preconditioners including jacobi, algebraic multigrid (AMG), incomplete LU (ILU) decomposition, parallel sparse approximate inverse (SAI). The research shows that the algebraic multigrid preconditioned GMRES converge quickly and the times of iteration for convergence of AMG and ILU preconditioners change with parallel scale. The AMG preconditione should be regarded as an efficient alternative to standard one-level preconditioners, such as ILU and SAI, for helmholtz problem in compressible atmosphere model. We believe that these results would be useful to researchers developing atmosphere model solvers and preconditioners as well as users seeking appropriate solvers for their own applications.