Numerical simulations of classical particle dynamics usually break energy conservation due to discretization of time. Is there any explicit numerical scheme that does not break energy conservation (without global rescaling of velocities)? Is it even possible for a particle systems with discretized time to have something like energy conservation law?
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Related: Is there something similar to Noether's theorem for discrete symmetries? – Qmechanic Apr 01 '21 at 00:04
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Yes. Symplectic integrators are designed to do exactly that. They are intended for solving Hamiltonian systems where the Hamiltonian is conserved.
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It looks great, but symplectic integrators seem to be solving a different problem: they preserve symplectic form. Does conservation of symplectic form imply conservation of energy? – Pavlo. B. Apr 01 '21 at 01:50
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I came to conclusion that conservation of symplectic form indeed likely to imply conservation of "some" energy. We could imagine a closed volume in phase space around an equilibrium point. If the energy was not conserved (for instance, increased), the volume would "blow up", which is not possible in symplectic integrator schemes. – Pavlo. B. Apr 01 '21 at 22:48