Optimizing locality for ODE solvers

Runge-Kutta methods are popular methods for the solution of systems of ordinary differential equations and are provided by many scientific libraries. The performance of Runge-Kutta methods does not only depend on the specific application problem to be solved but also on the characteristics of the target machine. For processors with memory hierarchy, the locality of data referencing pattern has a large impact on the efficiency of a program. In this paper, we describe program transformations for Runge-Kutta methods resulting in programs with improved locality behavior. The transformations are based on properties of the solution method but are independent from the specific application problem or the specific target machine, so that the resulting implementation is suitable as library function. We show that the locality improvement leads to performance gains on different target machines. We also demonstrate how the locality of memory references can be further increased by exploiting the dependence structure of the right hand side function of specific ordinary differential equations.

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