How do Loop Transformations Affect the Energy Consumption of Multi-Threaded Runge-Kutta Methods?

Runge-Kutta methods are widely used and popular solutions method for scientific simulations based on differential equations and, thus, their efficient execution is crucial for many applications. Today, also the energy consumption is getting more and more important for high performance computing. In this article, we investigate the performance and the energy consumption of Runge-Kutta methods solving systems of ordinary differential equations on recent Intel processors. Our specific interest is the study of different program versions of multithreaded Runge-Kutta methods which result from loop transformations within the nested loops over stage vectors and systems sizes. Four program versions of the Runge-Kutta method DOPRI5 are chosen and are applied to systems of ordinary differential equations with different workload. Experiments have been performed for different numbers of threads and the performance, power and energy consumption is reported and analyzed.