Galerkin variational integrators and modified symplectic Runge-Kutta methods

In this work, the equivalence of Galerkin variational integrators and Runge-Kutta methods is studied. The construction of Galerkin variational integrators relies on the approximation of the action in the variational principle based on a choice of a finite-dimensional function space and a numerical quadrature formula. While a particular class of Galerkin variational integrators is known to be equivalent to symplectic partitioned Runge-Kutta methods (Marsden, J. E. & West, M. (2001) Discrete mechanics and variational integrators. Acta Numer., 10, 357-514; Hairer, E., Lubich, C. & Wanner, G. (2002) Geometric numerical integration. Springer Series in Computational Mathematics, vol. 31. Berlin, Heidelberg, New York: Springer), this is not true for general Galerkin variational integrators. Based on the proof in Hairer et al. (2002, Geometric numerical integration. Springer Series in Computational Mathematics, vol. 31. Berlin, Heidelberg, New York: Springer), we derive modified Runge-Kutta methods which are shown to be equivalent to a new class of Galerkin variational integrators. This new class distinguishes, compared to the one introduced in Marsden & West (2001, Discrete mechanics and variational integrators. Acta Numer., 10, 357-514), only in the dimension of the finite-dimensional function space. We derive conditions under which both classes of variational integrator are identical. This provides a proof of the order of convergence for the newly introduced class of Galerkin variational integrators. The analytically derived results are demonstrated numerically by the Kepler problem.