Abstract
An approach is described to the numerical solution of order conditions for Runge–Kutta methods whose solutions evolve on a given manifold. This approach is based on least squares minimization using the Levenberg–Marquardt algorithm. Methods of order four and five are constructed and numerical experiments are presented which confirm that the derived methods have the expected order of accuracy.
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Jackiewicz, Z., Marthinsen, A. & Owren, B. Construction of Runge–Kutta methods of Crouch–Grossman type of high order. Advances in Computational Mathematics 13, 405–415 (2000). https://doi.org/10.1023/A:1016645730465
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DOI: https://doi.org/10.1023/A:1016645730465