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

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 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 S RKN 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 s9ABC6 and s11ABC6 (moderate accuracy), along with s17ABC8 and s19ABC8 (high accuracy) proved to be the most efficient schemes.are the 2N × 2N elements of the Hessian matrix D 2 H (x(t)) of the Hamiltonian function H computed on the phase space trajectory x(t). Eq. (3.5) is linear in w(t), with coefficients depending on the system's trajectory x(t). Therefore, one has to integrate the variational equations (3.5) along with the equations of motion (3.2), which means to evolve in time the general vector X(t) = (x(t), δx(t)) by solving the systemẊH (x(t)) · δx(t).( 3.7) In what follows we will briefly describe several numerical schemes for integrating the set of equations (3.7).

Nonlinear Topological Edge States: from Dynamic Delocalization to Thermalization

“…In the Lie formalism, the Hamilton equations of motion governing the evolution of an orbit starting at X(0), along with its variational equations, which govern the evolution of a small perturbation W (0) = δX(0) from this orbit [64,83,84] are…”

“…where Z = (X, δX), (˙) denotes the time derivative and L HV is a Lie operator whose general expression can, for example, be found in [64,84]. Therefore, the solution of the system's dynamical equations [Eq.…”

Thermalization in the one-dimensional Salerno model lattice

“…We solve Eq. (1) numerically by using an explicit Runge-Kutta algorithm of order 8, called DOP853 [42][43][44]. We set the relative energy error H(t)−H(0) H(0) and norm A(t)−A(0) A(0) error threshold 10 −4 .…”

On the Behavior of the Generalized Alignment Index (GALI) Method for Regular Motion in Multidimensional Hamiltonian Systems

“…Since Hamiltonian (1) can be split in two integrable parts H N ( x, p) = A( p) + B( x), with A( p) being the kinetic energy, which is a function of only the momenta p i , and B( x) being the potential energy depending only on the coordinates x i , we implement an efficient fourth-order symplectic integration scheme called ABA864 [24][25][26] for integrating the system's equations of motion. Symplectic integrators are numerical schemes specifically designed to preserve the symplectic structure of Hamiltonian systems.…”