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Mathematics of Computation

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Order barriers for continuous explicit Runge-Kutta methods

Authors: Brynjulf Owren and Marino Zennaro
Journal: Math. Comp. 56 (1991), 645-661
MSC: Primary 65L06
MathSciNet review: 1068811
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Abstract: In this paper we deal with continuous numerical methods for solving initial value problems for ordinary differential equations, the need for which occurs frequently in applications. Whereas most of the commonly used multi-step methods provide continuous extensions by means of an interpolant which is available without making extra function evaluations, this is not always the case for one-step methods. We consider the class of explicit Runge-Kutta methods and provide theorems used to obtain lower bounds for the number of stages required to construct methods of a given uniform order p. These bounds are similar to the Butcher barriers known for the discrete case, and are derived up to order $ p = 5$. As far as we know, the examples we present of 8-stage continuous Runge-Kutta methods of uniform order 5 are the first of their kind.

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Article copyright: © Copyright 1991 American Mathematical Society

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