Title: Structure-preserving ERKN Integrators

Prof. Xinyuan Wu
Department of Mathematics, Nanjing University, China

Abstract:
This talk introduces extended symplectic Runge-Kutta-Nystr\"om
(ESRKN) integrators for a system of oscillatory second-order
differential equations $q" + Mq = f(q)$ with a real symmetric
positive semi-definite matrix $M$, based on the multidimensional
extended Runge-Kutta-Nystr\"om (ERKN) methods proposed by Wu et al.
[ERKN integrators for systems of oscillatory second-order
differential equations, Comput. Phys. Comm. 181 (2010) 1873-1887].
Since the oscillatory system with $f(q) = -\nabla V(q)$ is equivalent
to a separable Hamiltonian system, the symplecticness conditions are
derived for the system with Hamiltonian
\[ H(p,q) = \frac{1}{2}p^Tp + \frac{1}{2}q^TMq + V(q). \]
The symplecticness conditions for ERKN methods generalize the
symplecticness conditions for classical Runge-Kutta-Nystr\"om (RKN)
methods. Furthermore, the energy-preserving conditions are presented
as well for the Hamiltonian system.