Energy conserving methods for Hamiltonian PDEs based on spectral space   decomposition
  In this paper we discuss energy conservation issues related to the numerical solution of the nonlinear wave equation, when a Fourier expansion is considered for the space discretization. The obtained semi-discrete problem is then solved in time by means of energy-conserving Runge-Kutta methods in the HBVMs class. 
Luigi Brugnano, Gianluca Frasca Caccia, Felice Iavernaro
math.NA