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IMA Journal of Numerical Analysis Advance Access originally published online on January 13, 2007
IMA Journal of Numerical Analysis 2007 27(4):798-817; doi:10.1093/imanum/drl040
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© The author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A family of A-stable Runge–Kutta collocation methods of higher order for initial-value problems

Jesús Vigo-Aguiar{dagger} and Higinio Ramos{ddagger}

Scientific Computing Group, Universidad de Salamanca, Salamanca, Spain

{dagger} Email: jvigo{at}usal.es

{ddagger} Corresponding author. Email: higra{at}usal.es

Received on 27 January 2006. Revised on 7 December 2006.


  Abstract

We consider the construction of a special family of Runge–Kutta (RK) collocation methods based on intra-step nodal points of Chebyshev–Gauss–Lobatto type, with A-stability and stiffly accurate characteristics. This feature with its inherent implicitness makes them suitable for solving stiff initial-value problems. In fact, the two simplest cases consist in the well-known trapezoidal rule and the fourth-order Runge–Kutta–Lobatto IIIA method. We will present here the coefficients up to eighth order, but we provide the formulas to obtain methods of higher order. When the number of stages is odd, we have considered a new strategy for changing the step size based on the use of a pair of methods: the given RK method and a linear multistep one. Some numerical experiments are considered in order to check the behaviour of the methods when applied to a variety of initial-value problems.

Key Words: Runge–Kutta collocation methods; initial-value problems; A-stability


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