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-order Runge–Kutta methods, providing a natural framework for the
comparison. The stability and accuracy properties of the methods are analyzed by
theoretical measures, and these are compared with the results of numerical
tests. In serial, the eighth-order pair of Prince and Dormand (DOP8) is most
efficient. But other high-order methods can be more efficient than DOP8 when
implemented in parallel. This is demonstrated by comparing a parallelized
version of the well-known ODEX code with the (serial) DOP853 code. For an
-body problem
with
,
the experimental extrapolation code is as fast as the tuned Runge–Kutta pair at
loose tolerances, and is up to two times as fast at tight tolerances.
Division of Computer, Electrical,
and Mathematical Sciences and Engineering
King Abdullah University of Science and Technology
Thuwal 23955-6900
Saudi Arabia