Potential of chaotic iterative solvers for CFD

Computational Fluid Dynamics (CFD) has enjoyed the speed-up available from supercomputer technology advancements for many years. In the coming decade, however, the architecture of supercomputers will change, and CFD codes must adapt to remain current. Based on the predictions of next-generation supercomputer architectures it is expected that the first computer capable of 1018 floating-point-operations-per-second (1 ExaFLOPS) will arrive in around 2020. Its architecture will be governed by electrical power limitations, whereas previously the main limitation was pure hardware speed. This has two significant repercussions. Firstly, due to physical power limitations of modern chips, core clock rates will decrease in favour of increasing concurrency. This trend can already been seen with the growth of accelerated “many-core” systems, which use graphics processing units (GPUs) or co-processors. Secondly, inter-nodal networks, typically using copper-wire or optical interconnect, must be reduced due to their proportionally large power consumption. This places more focus on shared-memory communications, with distributed-memory communication (predominantly MPI - “Message Passing Interface”) becoming less important. The current most powerful computer, Tianhe-2, capable of 33 PFlops, consists of 3,120,000 cores. The first exascale machine, which will be 30 times more powerful, is likely to be 300-times more parallel – which is a massive acceleration in parallelization compared to the last 50 years. This concurrency will come primarily from intra-node parallelization. Whereas Tianhe-2 features an already-large O(100) cores per node, an exascale machine must consist of O(1k-10k) cores per node. CFD has benefited from weak scalability (the ability to retain performance with a constant elements-per-core-ratio) for many years; its strong scalability (the ability to reduce the elements-per-core ratio) has been poor and mostly irrelevant. With the shift to massive parallelism in the next few years, the strong scalability of CFD codes must be investigated and improved. In this paper, a brief summary of earlier results is given, which identified the linear-equation system solver as one of the least-scalable parts of the code. Based on these results, a chaotic iterative solver, which is a totally-asynchronous, non-stationary, linear solver for high-scalability, is proposed. This paper focuses on the suitability of such a solver, by investigating the linear equation systems produced by typical CFD problems. If the results are optimistic, future work will be carried out to implement and test chaotic iterative solvers.