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

摘要

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 parallel implementations.We cast extrapolation and deferred correction methods as fixed-orderRunge–Kutta methods, providing a natural framework for the comparison. Thestability and accuracy properties of the methods are analyzed by theoreticalmeasures, and these are compared with the results of numerical tests. In serial,the eighth-order pair of Prince and Dormand (DOP8) is most efficient. Butother high-order methods can be more efficient than DOP8 when implemented inparallel. This is demonstrated by comparing a parallelized version of the wellknownODEX code with the (serial) DOP853 code. For an N-body problem withN = 400, the experimental extrapolation code is as fast as the tuned Runge–Kuttapair at loose tolerances, and is up to two times as fast at tight tolerances.