三维Euler方程组隐式LU分解的高性能并行计算

High performance parallel computing of implicit LU scheme for 3D Euler equations

许多非定常无粘流体力学问题的数值模拟都需要利用Euler方程组来进行计算,而由于在隐格式下,所选取的时间步长可以比在显格式下时大得多,所以隐格式越来越受到重视,其中隐式LU分解是最常用的方法之一。对三维Euler方程组,采用隐式LU分解进行计算时,网格点所在的各个对角阵面之间存在数据依赖关系,本文分析了采用区域分解且边界上用显格式代替隐格式进行计算的高效性,在长方体建筑物内的爆炸模拟表明,在有112个CPU的某MPP巨型机上,并行计算效率超过60%。本文还分析了计算结果与串行计算时的差异,以及利用区域重叠减小这种差异的方法,同时考虑了对处理器进行合理的逻辑组织,将计算网格映射到处理器网格,以最大限度减少通信开销的方法。文中最后以一个爆炸毁伤的例子实际说明了所述方法的可行性与高效性。

Euler equations play key roles in the simulation of many non-stationary non-viscid fluids. For the time-step selected is in general much larger than that in explicit ones, the implicit difference formulations have been focused on in these years. Among these formulations, the implicit LU scheme is one of the altematives used in many programs. While this scheme is selected, for 3D Euler equations discreted on rectangle-type grids, the computation of a diagonal face depends on the last for the L-part and on the next for the U-part. In this paper, a high performance parallel algorithm is designed based on the domain decomposition with overlap. On the outer-most interface, the implicit formulation is replaced by an explicit formulation. The results from the simulation of the explosion in a box shows that the parallel efficiency is satisfactory and can exceed 60% on a MPP with 112 CPUs. Secondly, the difference of the parallel results from the serial ones are analyzed and based on larger overlapping, there provides a scheme to decrease this difference. Finally, there analyzed the mappings of grid-points to processors and the best one is found to minimizing communication overhead. In the end of this paper, the provided schemes are proved by an example, which is the explosion in a box.