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Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions. (English) Zbl 1435.82050

Summary: 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 Schrödinger 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.

MSC:

82M37 Computational molecular dynamics in statistical mechanics
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
81V70 Many-body theory; quantum Hall effect
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65K10 Numerical optimization and variational techniques
35Q55 NLS equations (nonlinear Schrödinger equations)
34D08 Characteristic and Lyapunov exponents of ordinary differential equations

Software:

TIDES; Matlab
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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