Challenges in geometric numerical integration
Ernst Hairer
Abstract.
Geometric Numerical Integration is a subfield of the numerical treatment of differential
equations. It deals with the design and analysis of algorithms that preserve the structure of
the analytic flow. The present review discusses
numerical integrators, which nearly preserve the energy of Hamiltonian
systems over long times.
Backward error analysis gives important insight in the situation, where the product
of the step size with the highest frequency is small. Modulated Fourier expansions
permit to treat nonlinearly perturbed fast oscillators.
A big challenge that remains is to get insight into the long-time behavior
of numerical integrators for fully nonlinear oscillatory problems, where the product
of the step size with the highest frequency is not small.
Key Words.
Geometric numerical integration, energy preservation.