Abstract
A constructive characterization of the class of Runge–Kutta methods that satisfy the M-condition is presented. Such characterization describes a method in terms of a skew symmetric matrix \(\Sigma \) and the coefficient vector b instead of the standard Butcher parameters (A, b) and is based on the simplifying assumptions satisfied by the method and its associated quadrature formula. By using the new parameters we are able to classify easily all M-methods of a given number of stages according to their order and the simplifying conditions. The cases with \(s=2,3\) and 4 stages are presented.
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Notes
It is assumed that the quadrature formula is \(\int _0^1 g(t) \mathrm{d}t \simeq \sum _{i=1}^s b_i g(c_i)\).
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This work was supported by project MTM2013-47318-C2-1-P.
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Calvo, M., Montijano, J.I. & Rández, L. Constructive characterization of Runge–Kutta methods that satisfy the M-condition. SeMA 74, 345–359 (2017). https://doi.org/10.1007/s40324-017-0126-0
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DOI: https://doi.org/10.1007/s40324-017-0126-0