Performance Optimization of RK Methods Using Block-Based Pipelining

The efficiency of modern microprocessors is extremely sensitive towards the structure and memory access pattern of programs to be executed. This is caused by memory hierarchies which were introduced to reduce average memory access times. In this paper, we consider embedded Runge-Kutta (RK) methods for the solution of ordinary differential equations arising from space discretization problems for partial differential equations and study their efficient implementation on modern microprocessors. Different program variants with different execution orders and storage schemes are investigated. In particular, we explore how the potential parallelism in the stage vector computation can be exploited in a pipelining approach in order to improve the locality behavior of the RK implementations. Experiments show that this results in efficiency improvements on several recent processors.