Simpson's quadrature for a nonlinear variational symplectic scheme

TL;DR

This paper introduces a novel nonlinear symplectic numerical scheme based on Simpson's quadrature for integrating dynamical systems derived from the least action principle, demonstrating promising convergence in experiments.

Contribution

It presents a new variational symplectic method using Simpson's quadrature, enhancing numerical integration of nonlinear dynamical systems.

Findings

Good convergence observed in nonlinear pendulum simulations

Scheme preserves symplectic structure

Method applicable to systems from least action principle

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 between two time steps and we approximate the action in one time step by the Simpson's quadrature formula. The resulting scheme is nonlinear and symplectic. First numerical experiments concern a nonlinear pendulum and we have observed experimentally very good convergence properties.