Parallel Low-Storage Runge-Kutta Solvers for ODE Systems with Limited Access Distance

• Authors: Matthias Korch;Thomas Rauber
• Venue: International Journal of High Performance Computing Applications
• Year: 2011

We consider the solution of initial value problems (IVPs) of large systems of ordinary differential equations (ODEs) for which memory space requirements determine the choice of the integration method. In particular, we discuss the space-efficient sequential and parallel implementation of embedded Rungeâ聙聰Kutta (RK) methods. Our focus is on the exploitation of a special structure of commonly appearing ODE systems, referred to as â聙聵â聙聵limited access distance,â聙聶â聙聶 to improve scalability and memory usage. Such systems may arise, for example, from the semi-discretization of partial differential equations (PDEs). The storage space required by classical RK methods is directly proportional to the dimension n of the ODE system and the number of stages s of the method. We propose an implementation strategy based on a pipelined processing of the stages of the RK method and show how the memory usage of this 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 parallel implementation strategies in detailed runtime experiments on different modern parallel architectures.