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I'm interested in building some numerical integrators. Specifically symplectic integrators. Now, my Mathematica has been coming along, day by day, slowly but surely. Still not at a stage where I can dive right in and design my own integrators yet though, hence my timing of the question.

I do have some experience with other platforms like Maple and Matlab. In terms of designing integrators, Matlab is extremely easy. However, I don't believe it has the same computational power as Mathematica which is an issue for me.

More to the point, I know that there are symplectic integration schemes available in Mathematica using SymplecticPartitionedRungeKutta function under NDSolve. Also I know that there ImplicitRungeKutta which is also under NDSolve.

Some questions

  1. Under the SymplecticPartitionedRungeKutta regime of integrators is there also an option for implicit methods? The documentation is slightly unclear. It appears that they are designed for separable Hamiltonians, or in other words, explicit methods.
  2. In terms of actually designing my own integrators, say to compare with the algorithms supplied my Wolfram what are the available resources that the community is aware of? I have found some very comprehensive well written tutorials on all areas associated with Mathematica. However, with regards to numerical integration, it seems to be more geared towards applications.
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