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

General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs