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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Continuous explicit Runge-Kutta methods of order $5$
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by J. H. Verner and M. Zennaro PDF
Math. Comp. 64 (1995), 1123-1146 Request permission


A continuous explicit Runge-Kutta (CERK) method provides a continuous approximation to an initial value problem. Such a method may be obtained by appending additional stages to a discrete method, or alternatively by solving the appropriate order conditions directly. Owren and Zennaro have shown for order 5 that the latter approach yields some CERK methods that require fewer derivative evaluations than methods obtained by appending stages. In contrast, continuous methods of order 6 that require the minimum number of stages can be obtained by appending additional stages to certain discrete methods. This article begins a study to understand why this occurs. By making no assumptions to simplify solution of the order conditions, the existence of other types of CERK methods of order 5 is established. While methods of the new families may not be as good for implementation as the Owren-Zennaro methods, the structure is expected to lead to a better understanding of how to construct families of methods of higher order.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Math. Comp. 64 (1995), 1123-1146
  • MSC: Primary 65L06; Secondary 65Y20
  • DOI:
  • MathSciNet review: 1284672