In this volume, we present a pedagogic overview of the freedom one has in choosing the coefficients in explicit Runge-Kutta integration schemes. We limit ourselves to the simple cases in which there are at most two new evaluations of the right-hand side of the differential equation per time step. We start with first-order differential equations, to explain the general procedures, but then we limit ourselves to the type of second-order differential equation that occurs in classical mechanics, where the forces are dependent only on positions, and independent of the velocities. In that case we derive some of the classical schemes, and generalize this to include a first evaluation that does not take place at the beginning of a time step. We show how such a generalized approach naturally leads us to the scheme, as a particular form of a generalized explicit Runge-Kutta scheme.

We thank Kristin Cordwell and xxx for their comments on the manuscript.

Piet Hut, Jun Makino, and Douglas Heggie

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