Communications in Applied Mathematics and Computational Science

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

David Ketcheson and Umair bin Waheed

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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 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.

Article information

Commun. Appl. Math. Comput. Sci., Volume 9, Number 2 (2014), 175-200.

Received: 4 November 2013
Revised: 4 May 2014
Accepted: 8 May 2014
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65L06: Multistep, Runge-Kutta and extrapolation methods
Secondary: 65Y05: Parallel computation

Runge–Kutta methods extrapolation deferred correction ordinary differential equations high-order methods parallel


Ketcheson, David; bin Waheed, Umair. A comparison of high-order explicit Runge–Kutta, extrapolation, and deferred correction methods in serial and parallel. Commun. Appl. Math. Comput. Sci. 9 (2014), no. 2, 175--200. doi:10.2140/camcos.2014.9.175.

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