Prolongation–collocation variational integrators

We introduce a novel technique for constructing higher-order variational integrators for Hamiltonian systems of ordinary differential equations. In the construction of the discrete Lagrangian we adopt Hermite interpolation polynomials and the Euler–Maclaurin quadrature formula and apply collocation to the Euler–Lagrange equation and its prolongation. Considerable attention is devoted to the order analysis of the resulting variational integrators in terms of approximation properties of the Hermite polynomials and quadrature errors. In particular, the order of the variational integrator can be computed a priori based on the quadrature error estimate. The analysis in the paper is straightforward compared to the order theory for Runge–Kutta methods. Finally, a performance comparison is presented on a selection of these integrators.