Lagrangian and Hamiltonian Taylor Variational Integrators
This work provides a more accurate and efficient numerical method for simulating mechanical systems, benefiting computational physics and engineering applications.
The authors develop a variational integrator using Taylor's method to approximate the Euler-Lagrange boundary-value problem, achieving one order higher accuracy than comparable shooting methods and enabling efficient quadrature evaluations. Numerical experiments demonstrate its efficacy and efficiency.
In this paper, we present a variational integrator that is based on an approximation of the Euler--Lagrange boundary-value problem via Taylor's method. This can viewed as a special case of the shooting-based variational integrator. The Taylor variational integrator exploits the structure of the Taylor method, which results in a shooting method that is one order higher compared to other shooting methods based on a one-step method of the same order. In addition, this method can generate quadrature nodal evaluations at the cost of a polynomial evaluation, which may increase its efficiency relative to other shooting-based variational integrators. A symmetric version of the method is proposed, and numerical experiments are conducted to exhibit the efficacy and efficiency of the method.