基于并行可扩展科学计算工具集求解GRAPES全球非静力模式亥姆霍兹问题  

Solving Helmholtz equations of the GRAPES global nonhydrostatic model based on the PETSc

作　　者：张理论[1] 宋君强[1] 赵文涛[1] 胡江林[2] 

机构地区：[1]国防科学技术大学计算机学院软件研究所,长沙410073 [2]中国气象局数值预报中心,北京100081

出　　处：《气象学报》2011年第3期432-439,共8页Acta Meteorologica Sinica

基　　金：国家自然科学基金项目(40505023)

摘　　要：亥姆霍兹方程是非静力平衡大气模式动力内核的主要计算瓶颈之一,其离散矩阵性态差,采用常见预条件Krylov迭代往往收敛很慢。随着全球非静力平衡大气模式时空分辨率的不断提高,亥姆霍兹方程求解面临求解精度和计算时间的双重困难。在高分辨率情况下,迭代计算步数和计算量剧增,而且很多传统的预条件迭代求解方法不收敛,迫切需要研究收敛性和并行可扩展性兼备的预条件迭代方法。为此,在安腾机群上建立了基于并行可扩展科学计算工具集(PETSc)的GRAPES全球非静力平衡模式亥姆霍兹问题并行解法器对比研究平台,结合高性能预条件库(hypre),完成了对GRAPES在用解法器、代数多重网格、并行不完全LU分解(EUCLID)及加性Schwarz区域分解等一系列克雷洛夫预条件迭代的分析对比。结果显示PETSc结合高性能预条件库的并行方案是解决GRAPES模式三维亥姆霍兹方程高效并行计算的一个有效途径,其中以代数多重网格预条件迭代的性能最突出;并行加速比分析显示,代数多重网格预条件迭代的并行可扩展性明显优于GRAPES现有解法器,更适用于更高精度和较大规模并行计算。3D Helmholtz equations are the key bottleneck of the dynamic core of nonhydrostatic atmosphere model. These equa- tions are complicated to solve with poor-conditioned linear systems. Common preconditioners with Krylov iterative methods al- ways converge slowly. With the increase of time and space resolution of the global nonhydrostatic model, solving Helmholtz e- quations are more stubborn in both precision and efficiency. Under high resolution, both number of iteration step and complexi- ty per step aggravate sharply, some traditional preconditioned Krylov methods ever do not converge. It is important to find a method for solving Helmholtz equations with both good convergence characteristic and high parallel scalability. To investigate efficient solvers, a common platform for comparing algorithm should be constructed beforehand. On the Itanium2 cluster, the PETSc and the hypre are seamlessly linked together as a parallel problem solving environment of the GRAPES global model. The LU preconditioner used in the GRAPES, the algebraic mulitgrid （AMG） preconditioner, the parallel LU preconditioner （EUCLID） and Schwarz domain decomposition preconditioners are compared. This research shows that the combination of the PETSc and the hypre is effective for computing optimization of Helmholtz equations. Especially the case of AMG （Boomer- AMG） is the most efficient. The analyses of the parallel speedup show that the AMG preconditioned iterative is remarkablely scalable, thus is fitter for massively parallel computing.

关 键 词：GRAPES 亥姆霍兹方程 并行可扩展科学计算工具集 高性能预条件库 预条件 

分 类 号：P456.7[天文地球—大气科学及气象学]