A symplectic integrator is a numerical discretization scheme for solving Hamilton's equations which takes into account the symplectic structure and, in particular, the conservation laws, at the discretization level, thus resulting in better long-time behaviour of numerical solutions than that of generic discretization schemes. There are analogues for classical field theory, which take into account the resulting multisymplectic structure.
wikipedia symplectic integrator, multisymplectic integrator
Denis Donnelly, Edwin Rogers, Symplectic integrators: An introduction, Am. J. Phys. 73, 938 (2005) doi
symplectic numerical integration at DAMTP
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Sebastian Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal. 36, 1549–1570, 1996 citeseer
Y. B. Suris, Hamiltonian Runge-Kutta type methods and their variational formulation (1990)
The idea can be adapted to dissipative systems as well:
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