Hamiltonian systems possess several important structures and conservation laws: most notably a symplectic or Poisson structure and conservation of energy, momentum maps and Casimir invariants. Numerical algorithms which preserve these structures usually show greatly reduced errors compared to algorithms that do not preserve these structures, as well as much better long-time stability.
In this lecture, important mathematical structures of Hamiltonian systems will be reviewed and consequences of their non-preservation in numerical simulations higlighted. Some basic structure-preserving algorithms for canonical Hamiltonian systems will be introduced and compared with their non-structure-preserving counterparts. Finally, and outlook will be given on the structure-preserving discretisation of noncanonical Hamiltonian systems like those found in fluid dynamics and plasma physics. For such systems, there are no standard methods available like the many that are known for canonical Hamiltonian systems and the development of new methods is much more challenging.