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Implications of order reduction for implicit Runge-Kutta methods

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Abstract

Stability analysis of Runge-Kutta (RK) formulas was originally limited to linear ordinary differential equations (ODEs). More recently such analysis has been extended to include the behaviour of solutions to nonlinear problems. This extension led to additional stability requirements for RK methods. Although the class of problems has been widened, the analysis is still restricted to a fixed stepsize. In the case of differential algebraic equations (DAEs), additional order conditions must be satisfied [6] to achieve full classical ODE order and avoid possible “order reduction”. In this case too, a fixed stepsize analysis is employed. Such analysis may be of only limited use in quantifying the effectiveness of adaptive methods on stiff problems.

In this paper we examine the phenomenon of “order reduction” and its implications on variable-step algorithms. We introduce a global measure of order referred to here as the “observed order” which is based on the average stepsize over the region of integration. This measure may be better suited to the study of stiff systems, where the stepsize selection algorithm will vary the stepsize considerably over the interval of integration. Observed order gives a better indication of the relationship between accuracy and cost. Using this measure, the “observed order reduction” will be seen to be less severe than that predicated by fixed stepsize order analysis.

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Authors and Affiliations

  1. Department of Computer Science, University of Toronto, M5S 1A4, Canada

    Craig Mac Donald & Wayne H. Enright

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  1. Craig Mac Donald
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  2. Wayne H. Enright
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Communicated by Å. Björck

Supported by the Information Technology Research Centre of Ontario, and the Natural Science and Engineering Research Council of Canada.

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Donald, C.M., Enright, W.H. Implications of order reduction for implicit Runge-Kutta methods. Numer Algor 2, 351–369 (1992). https://doi.org/10.1007/BF02139474

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  • DOI: https://doi.org/10.1007/BF02139474

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