Abstract
General linear methods for the solution of ordinary differential equations are both multivalue and multistage. The order conditions will be stated and analyzed using a B-series approach. However, imposing the G-symplectic structure modifies the nature of the order conditions considerably. For Runge–Kutta methods, rooted trees belonging to the same tree have equivalent order conditions; if the trees are superfluous, they are automatically satisfied and can be ignored. For G-symplectic methods, similar results apply but with a more general interpretation. In the multivalue case, starting conditions are a natural aspect of the meaning of order; unlike the Runge–Kutta case for which “effective order” or “processing” or “conjugacy” has to be seen as having an artificial meaning. It is shown that G-symplectic methods with order 4 can be constructed with relatively few stages, \(s=3\), and with only \(r=2\) inputs to a step.
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The authors acknowledge the support of the Marsden Fund. Furthermore, they thank the referees for their valuable comments.
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Communicated by Anne Kværnø.
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Butcher, J.C., Imran, G. Order conditions for G-symplectic methods. Bit Numer Math 55, 927–948 (2015). https://doi.org/10.1007/s10543-014-0541-x
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DOI: https://doi.org/10.1007/s10543-014-0541-x