三维热传导方程的Krylov子空间方法并行分析

热传导方程在地下水流动数值模拟、油藏数值模拟等工程计算中有着广泛应用，其并行实现是加速问题求解速度、提高问题求解规模的重要手段，因此热传导方程的并行求解具有重要意义。对Krylov子空间方法中的CG和GMRES算法进行并行分析，并对不同的预处理CG算法作了比较。在Linux集群系统上，以三维热传导模型为例进行了数值实验。实验结果表明，CG算法比GMRES算法更适合建立三维热传导模型的并行求解。此外，CG算法与BJACOBI预条件子的整合在求解该热传导模型时，其并行程序具有良好的加速比和效率。因此，采用BJACOBI预处理技术的CG算法是一种较好的求解三维热传导模型的并行方案。

Analysis of parallel Krylov subspace method for three-dimensional heat equation

Heat equation has been widely used in engineering, such as numerical simulation of groundwater flow, reservoir simulation and so on. The parallelism of heat equation is an important means of accelerating the simulation process and enhancing the modeling capabilities. This paper analyzed the parallelism of GMRES and CG algorithm included in Krylov subspace method, made a comparison with different preconditioned conjugate gradient methods. Numerical experiments on the three-dimensional heat equation were carried out on Linux clusters. The numerical results demonstrate that CG algorithm is more suitable than the GMRES algorithm for parallelizing the three-dimensional heat equation. The parallel program has a desirable speedup and efficiency when use CG algorithm integrating with BJACOBI preconditioner to solve the three-dimensional heat equation. So a better parallel solution to the three-dimensional heat equation is CG algorithm integrating with BJACOBI preconditioner.