A variational symplectic scheme based on Simpson's quadrature

TL;DR

This paper introduces a new explicit symplectic numerical method for dynamical systems based on variational principles and Simpson's quadrature, demonstrating stability and energy conservation through numerical tests.

Contribution

It presents a novel variational symplectic scheme using Simpson's quadrature for time integration, combining explicitness, stability, and energy conservation.

Findings

The scheme is explicit and symplectic.

It conserves an approximate energy.

Numerical tests confirm theoretical properties.

Abstract

We propose a variational symplectic numerical method for the time integration of dynamical systems issued from the least action principle. We assume a quadratic internal interpolation of the state and we approximate the action in a small time step by the Simpson's quadrature formula. The resulting scheme is explicited for an elementary harmonic oscillator. It is a stable, explicit, and symplectic scheme satisfying the conservation of an approximate energy. Numerical tests illustrate our theoretical study.