Accuracy of classical conservation laws for Hamiltonian PDEs under Runge-Kutta discretizations

We investigate conservative properties of Runge-Kutta methods for Hamiltonian PDEs. It is shown that multi-symplecitic Runge-Kutta methods preserve precisely norm square conservation law. Based on the study of accuracy of Runge-Kutta methods applied to ordinary and partial differential equations, we present some results on the numerical accuracy of conservation laws of energy and momentum for Hamiltonian PDEs under Runge-Kutta discretizations.