  In this paper we propose and investigate a general approach to constructing
local energy-preserving algorithms which can be of arbitrarily high order in
time for solving Hamiltonian PDEs. This approach is based on the temporal
discretization using continuous Runge-Kutta-type methods, and the spatial
discretization using pseudospectral methods or Gauss--Legendre collocation
methods. The local energy conservation law of our new schemes is analyzed in
detail. The effectiveness of the novel local energy-preserving integrators is
demonstrated by coupled nonlinear Schr\"odinger equations and 2D nonlinear
Schr\"odinger equations with external fields. Our new schemes are compared with
some classical multi-symplectic and symplectic schemes in numerical
experiments. The numerical results show the remarkable \emph{long-term}
behaviour of our new schemes.
