Diagonal implicit symplectic ERKN methods for solving oscillatory Hamiltonian systems

TL;DR

This paper develops diagonal implicit symplectic ERKN methods tailored for oscillatory Hamiltonian systems, demonstrating their stability and superior numerical performance through theoretical analysis and numerical experiments.

Contribution

It introduces new diagonal implicit symplectic ERKN methods specifically designed for oscillatory Hamiltonian systems, with stability analysis and numerical validation.

Findings

Methods exhibit excellent stability properties.

Numerical experiments confirm high accuracy and efficiency.

Significant improvement over existing methods.

Abstract

This paper studies diagonal implicit symplectic extended Runge--Kutta--Nystr\"{o}m (ERKN) methods for solving the oscillatory Hamiltonian system $H(q,p)=\dfrac{1}{2}p^{T}p+\dfrac{1}{2}q^{T}Mq+U(q)$. Based on symplectic conditions and order conditions, we construct some diagonal implicit symplectic ERKN methods. The stability of the obtained methods is discussed. Three numerical experiments are carried out and the numerical results demonstrate the remarkable numerical behavior of the new diagonal implicit symplectic methods when applied to the oscillatory Hamiltonian system.