Vol. 9, No. 2, 2014

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

David I. Ketcheson and Umair bin Waheed

Vol. 9 (2014), No. 2, 175–200

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 N-body problem with N = 400, 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.

Runge–Kutta methods, extrapolation, deferred correction, ordinary differential equations, high-order methods, parallel
Mathematical Subject Classification 2010
Primary: 65L06
Secondary: 65Y05
Received: 4 November 2013
Revised: 4 May 2014
Accepted: 8 May 2014
Published: 13 June 2014
David I. Ketcheson
Division of Computer, Electrical, and Mathematical Sciences and Engineering
King Abdullah University of Science and Technology
Thuwal 23955-6900
Saudi Arabia
Umair bin Waheed
Division of Physical Sciences and Engineering
King Abdullah University of Science and Technology
Thuwal 23955-6900
Saudi Arabia