# Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions

## TL;DR

This paper compares the computational efficiency of various numerical integration methods, especially symplectic integrators, for simulating the dynamics of many-body Hamiltonian systems in one and two dimensions, focusing on accuracy and energy conservation.

## Contribution

It provides a comprehensive comparison of different numerical schemes, including explicit symplectic integrators, for classical many-body models like FPUT and DDNLS, highlighting the most efficient methods.

## Key findings

- Symplectic integrators ABA864 and SRKN^a_{14} perform best for the FPUT chain.
- For DDNLS models, s9ABC6, s11ABC6, s17ABC8, and s19ABC8 are the most efficient schemes.
- Symplectic methods effectively preserve energy and accurately reproduce system dynamics.

## Abstract

We investigate the computational performance of various numerical methods for the integration of the equations of motion and the variational equations for some typical classical many-body models of condensed matter physics: the Fermi-Pasta-Ulam-Tsingou (FPUT) chain and the one- and two-dimensional disordered, discrete nonlinear Schr\"odinger equations (DDNLS). In our analysis we consider methods based on Taylor series expansion, Runge-Kutta discretization and symplectic transformations. The latter have the ability to exactly preserve the symplectic structure of Hamiltonian systems, which results in keeping bounded the error of the system's computed total energy. We perform extensive numerical simulations for several initial conditions of the studied models and compare the numerical efficiency of the used integrators by testing their ability to accurately reproduce characteristics of the systems' dynamics and quantify their chaoticity through the computation of the maximum Lyapunov exponent. We also report the expressions of the implemented symplectic schemes and provide the explicit forms of the used differential operators. Among the tested numerical schemes the symplectic integrators $ABA864$ and $SRKN^a_{14}$ exhibit the best performance, respectively for moderate and high accuracy levels in the case of the FPUT chain, while for the DDNLS models $s9\mathcal{ABC}6$ and $s11\mathcal{ABC}6$ (moderate accuracy), along with $s17\mathcal{ABC}8$ and $s19\mathcal{ABC}8$ (high accuracy) proved to be the most efficient schemes.