High-order Symmetric Schemes for the Energy Conservation of Polynomial Hamiltonian Problems

We define a class of arbitrary high order symmetric one-step methods that, when applied to Hamiltonian systems, are capable of precisely conserving the Hamiltonian function when this is a polynomial, whatever the initial condition and the stepsize h used. The key idea to devise such methods is the use of the so called discrete line integral, the discrete counterpart of the line integral in conservative vector fields. This approach naturally suggests a formulation of such methods in terms of block Boundary Value Methods, although they can be recast as Runge-Kutta methods, if preferred. c © 2009 European Society of Computational Methods in Sciences and Engineering