A variational symplectic scheme based on Lobatto's quadrature

Abstract

We present a variational integrator based on the Lobatto quadrature for the time integration of dynamical systems issued from the least action principle. This numerical method uses a cubic interpolation of the states and the action is approximated at each time step by Lobatto's formula. Numerical analysis is performed on both a harmonic oscillator and a nonlinear pendulum. The geometric scheme is conditionally stable, sixth-order accurate, and symplectic. It preserves an approximate energy quantity. Simulation results illustrate the performance and the superconvergence of the proposed method.

Figures (4)

• Figure 1: Harmonic oscillator evolution for the momentum $\, p \,$ and state $\, q$. Comparison of the Lobatto symplectic scheme against the exact solution for $N = 3$ meshes. Lobatto's solutions are very close to the exact ones. Momentum data have been rescaled.
• Figure 2: Over $1.0e5$ periods, the $\ell^\infty$ energy error growth rate is of: 2.5e-11 when $\, h=0.1$, 5.45e-14 when $\, h=0.05$, and 1.35e-13 when $\, h=0.025$.
• Figure 3: Nonlinear pendulum evolution for the momentum $\, p \,$ and state $\, q$. Comparison of the Lobatto symplectic scheme against the exact solution for $N = 5$ meshes over 5 periods. Lobatto's solutions remain very close to the exact ones. Note that the momentum data have been rescaled.
• Figure 4: Over $5.0e3$ periods, the $\ell^\infty$ energy error growth rate is of: 7.99e-13 when $\, h=0.1$, 5.39e-11 when $\, h=0.05$, and 5.33e-15 when $\, h=0.025$.