Automatic Generation of High-Order Finite-Difference Code with Temporal Blocking For Extreme-Scale Many-Core Systems

In this paper we describe the basic idea, implementation and achieved performance of our DSL for stencil computation, Formura, on systems based on PEZY-SC2 many-core processor. Formura generates, from high-level description of the differential equation and simple description of finite-difference stencil, the entire simulation code with MPI parallelization with overlapped communication and calculation, advanced temporal blocking and parallelization for many-core processors. Achieved performance is 4.78 PF, or 21.5% of the theoretical peak performance for an explicit scheme for compressive CFD, with the accuracy of fourth-order in space and third-order in time. For a slightly modified implementation of the same scheme, efficiency was slightly lower (17.5%) but actual calculation time per one timestep was faster by 25%. Temporal blocking improved the performance by up to 70%. Even though the B/F number of PEZY-SC2 is low, around 0.02, we have achieved the efficiency comparable to those of highly optimized CFD codes on machines with much higher memory bandwidth such as K computer. We have demonstrated that automatic generation of the code with temporal blocking is a quite effective way to make use of very large-scale machines with low memory bandwidth for large-scale CFD calculations.