nLab symplectic integrator

Idea

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.

References

• 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

• Daniel W. Markiewicz, Survey on symplectic integrators, pdf

• Robert McLachlan, Klas Modin, Olivier Verdier, Collective Lie–Poisson integrators on $\mathbb{R}^{3}$, arXiv:1307.2387

• A. Lew, J. E. Marsden, M. Ortiz, M. West, An overview of variational integrators, In L. P. Franca (ed.), Finite Element Methods: 70’s and Beyond. Barcelona (2003).

• Jerrold E. Marsden, George W. Patrick, Steve Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Commun. Math. Physics 199:2 (1998) 351-395 math.DG/9807080, doi

• François Demoures, François Gay-Balmaz, Tudor S. Ratiu, Multisymplectic variational integrators and space/time symplecticity, arXiv:1310.4772

• Ernst Hairer, Backward analysis of numerical integrators and symplectic methods, ps

• 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:

• Guilherme França, Michael I. Jordan, René Vidal, On dissipative symplectic integration with applications to gradient-based optimization, arxiv/2004.06840

category: applications