Scalability of Time- and Space-Efficient Embedded Runge-Kutta Solvers for Distributed Address Space

Embedded Runge-Kutta methods are well-known and efficient solution methods for initial value problems of ordinary differential equations (ODEs). In this paper, we discuss the parallel implementation of embedded Runge-Kutta methods for distributed address space. Our focus lies on the exploitation of a special structure of commonly appearing ODE systems to improve scalability and memory usage. We show how the memory space of a pipeline-like computation scheme can be reduced to less than three storage registers by an overlapping of vectors without compromising the choice of method coefficients or the potential for efficient stepsize control. We analyze and compare the scalability of different implementation strategies in detailed runtime experiments on different modern parallel architectures. These experiments show that our approach leads to a good scalability behavior even on large numbers of processors.