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I'm doing a gravitational n-body simulator and I'm thinking of implementing linear multistep methods like Adam-Bashforth. But is there any symplectic multistep methods?

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Yes, but you have to mean symplectic on a higher-dimensional phase space than your original problem that includes previous steps too. As I understand there are also some subtle stability issues too. Rather than try to summarize, I'll just refer you to chapter 15, section 4 of Hairer, Lubich, Wanner, Geometric Numerical Integration. That book is a must-have if you're working on these sorts of problems, albeit a little dense.

I also think it's telling that symplectic Runge-Kutta methods are always discussed front-and-center, whereas symplectic multi-step methods are barely mentioned in this book and not mentioned at all in this one. If you wanted the most payoff for the least effort you might have a better time implementing higher-order symplectic Runge Kutta methods instead.

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    $\begingroup$ "If you wanted the most payoff for the least effort you might have a better time implementing higher-order symplectic Runge Kutta methods instead.". Indeed, there are a few results that are discussed in the books about multistep methods, and generally it just shows that the possibilities are limiting. This is definitely a case where RK is more natural. $\endgroup$ – Chris Rackauckas Jun 4 at 19:44

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