Spectral Variational Integrators
For computational mechanics, this provides a high-accuracy, structure-preserving integration method that overcomes the order limitations of traditional variational integrators.
This paper introduces spectral variational integrators for Lagrangian mechanics that achieve geometric convergence while preserving symplectic and momentum properties, with optimal convergence rates proven theoretically and demonstrated numerically.
In this paper, we present a new variational integrator for problems in Lagrangian mechanics. Using techniques from Galerkin variational integrators, we construct a scheme for numerical integration that converges geometrically, and is symplectic and momentum preserving. Furthermore, we prove that under appropriate assumptions, variational integrators constructed using Galerkin techniques will yield numerical methods that are in a certain sense optimal, converging at the same rate as the best possible approximation in a certain function space. We further prove that certain geometric invariants also converge at an optimal rate, and that the error associated with these geometric invariants is independent of the number of steps taken. We close with several numerical examples that demonstrate the predicted rates of convergence.