Research On Parallel Algorithms Of The Fast Algorithm For Spherical Harmonic Expansions

The continuous progress of Numerical Weather Pridiction (NWP) model is promoted by the rapic development of computer technology, which makes more consummate and accurate the numerical method of discretization of continuous partial differential equations (PDE), and therefore, decreases the error of numerical solutions of PDE. Spectral transform method means using spherical harmonic expansions (SHE) with limited truncations to approximate some variable of a specific vertical layer of sphere surface, taking advantage of appropriate transformation of the variable between physical and spectral space to gain accurate solutions and decrease the computation, which mainly attributes to the non-linear terms of PDE. The physical space is formed by Cartesian products of the variable on the points of longitudinal-latitudinal grid, while the spectral space is formed by the corresponding coefficients of truncated SHE of the variable. Spectral model is advanced with its high accuracy and stability, while with the drawback of large mass of computation and storage.As a mathematical method, SHE is implemented in many domains, such as NWP model, Geophysics, Chemical physics, numerical solutions of PDE, etc. Spherical harmonics are the eigenfunctions of latitudinal differential operators and Laplace operators on sphere surface. Based on spherical harmonics, the spectral transform is the kernel of the computation of spectral model, consisting of Fourier Transform and Legendre Transform, which is applied in longitudinal and latitudinal direction respectively. Based on Rokhlin-Tygert's fast algorithms for SHE (RT Algorithm), this dissertation deeply investigates the parallelization of the algorithm, mainly including:(1) Analyze the characteristics of the spectral model of NWP and spherical harmonics, and study the theories and techniques of parallel computing, including MPI parallelism, CUDA parallelism, MPI+CUDA heterogeneous parallelism;(2) Deeply study and consummate Rokhlin-Tygert's fast algorithms for SHE, to which adding a procedure for computing the coefficients of 0-order normalized Legendre functions;(3) Design RT parallel algorithm according to the fast algorithms for SHE;(4) Implement MPI parallel program for RT Algorithm, which was tested on the 5-Terascale Clusters and Tianhe-1A supercomputer respectively, the results of which show that, when the wave number of triangular truncations M ? 1023, the parallel efficiency of the program would reach over 87%;(5) Implement MPI+CUDA heterogeneous parallel program for RT Algorithm, using CULA library functions. The result of the test on Tianhe-1A shows that, firstly, CUDA acceleration would be effective gradually when M ? 1023; secondly, the parallel efficiency of 2-process-per-node distribution would increase as M grows, and eventually transcend that of 1-process-per-node distribution; thirdly, when M ? 4095, the speedup of using 1024 processes on 512 nodes could reach over 6700.