GRAPES模式中Helmhothz方程两种求解方法的对比研究

A Study of the Two Helmholtz Solvers in the GRAPES Model Using GCR and GMRES

GRAPES是中国气象局自主研发的一个全球/区域分析预报系统。其模式计算方程组经过离散化之后,积分求解过程最终归结为对一个椭圆方程或Helmholtz(赫姆霍兹)方程的求解,这个求解是整个动力框架计算的核心。在目前GRAPES全球模式的准业务计算中,对于分辨率为0.5o的系统,Helmholtz方程的求解时间占到了整个模式计算时间的三分之一强。而且随着未来高分辨率模式的进一步加细,以及模式计算精度的提高,方程求解计算总量更是呈指数式增长。为此,本文分析了GRAPES模式中求解Helmholtz方程所采用的广义共轭余差法(GCR),并对比给出了利用PETSC函数库中提供的GMRES方法求解Helmholtz方程的一些初步测试结果。结果表明,采用高精度的GMRES方法可以减少模式预报偏差,改善模式预报准确度,在大规模并行计算时具有更好的可扩展性能。

GRAPES（Global and Regional Assimilation and PrEdiction System）is a new generation of NWP model in CMA （China Meteorological Administration） for the operational implementation. After the discretization of computing equations for GRAPES＇s model, the first calculation becomes the solution of the Helmholtz equations which is the kernel computing of the dynamic framework. The running time for solving the Helmholtz equations is more than one-third of the total cost for GRAPES-global mode at 0.5°x0.5° horizontal resolution with 38 vertical levels, and for the higher resolution model, the timecost is an exponential growth. The generalized conjugate residual method is employed to solve the 3D Helmholtz equation in the version of the GRAPES mode currently, as a contrast, another method which is based on GMRES（generalized minimal residual method）of PETSc（Portable, Extensible Toolkit for Scientific computation） is used here. The computation shows that the GMRES method with high precision can improve the forecast accuracy and has much better scalability for large-scale parallel computing.