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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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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