Construction and analysis of higher order Galerkin variational integrators
For researchers in computational mechanics and geometric numerical integration, this provides a systematic construction of higher-order variational integrators with proven stability and efficiency gains.
This work derives and analyzes higher-order variational integrators for structure-preserving simulation of mechanical systems, achieving increased accuracy and reduced computational cost while preserving geometric properties. Numerical tests demonstrate improved convergence rates compared to prior methods.
In this work we derive and analyze variational integrators of higher order for the structure-preserving simulation of mechanical systems. The construction is based on a space of polynomials together with Gauss and Lobatto quadrature rules to approximate the relevant integrals in the variational principle. The use of higher order schemes increases the accuracy of the discrete solution and thereby decrease the computational cost while the preservation properties of the scheme are still guaranteed. The order of convergence of the resulting variational integrators are investigated numerically and it is discussed which combination of space of polynomials and quadrature rules provide optimal convergence rates. For particular integrators the order can be increased compared to the Galerkin variational integrators previously introduced in Marsden & West 2001. Furthermore, linear stability properties, time reversibility, structure-preserving properties as well as efficiency for the constructed variational integrators are investigated and demonstrated by numerical examples.