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We compare the three main types of high-order one-step initial value solvers:extrapolation, spectral deferred correction, and embedded Runge--Kutta pairs.We consider orders four through twelve, including both serial and parallelimplementations. We cast extrapolation and deferred correction methods asfixed-order Runge--Kutta methods, providing a natural framework for thecomparison. The stability and accuracy properties of the methods are analyzedby theoretical measures, and these are compared with the results of numericaltests. In serial, the 8th-order pair of Prince and Dormand (DOP8) is mostefficient. But other high order methods can be more efficient than DOP8 whenimplemented in parallel. This is demonstrated by comparing a parallelizedversion of the well-known ODEX code with the (serial) DOP853 code. For an$N$-body problem with $N=400$, the experimental extrapolation code is as fastas the tuned Runge--Kutta pair at loose tolerances, and is up to two times asfast at tight tolerances.

A comparison of high order explicit Runge-Kutta, extrapolation, and deferred correction methods in serial and parallel