6
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Are there high order energy-conserving or symplectic methods for solving $y'=f(y)$?

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9
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There are the geometric integrations written by Ernst Hairer & co:

  • E. Hairer, C. Lubich and G. Wanner (2002): Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential equations. Springer Series in Comput. Math., vol. 31

  • E. Hairer and M. Hairer (2002): GniCodes - Matlab programs for geometric numerical integration

for which you can download the routines in FORTRAN77, Matlab or C++ from Professor Hairer's website. The Implicit Runge-Kutta (IRK) scheme he has implemented has order 8.

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