On the symplectic integration of the Klein Gordon lattice model

Bob Senyange

arXiv:1703.01328·math.DS·March 7, 2017

On the symplectic integration of the Klein Gordon lattice model

Bob Senyange

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TL;DR

This paper evaluates various symplectic integration methods for simulating the Klein Gordon lattice, focusing on efficiency and accuracy in weak chaos regimes through extensive numerical experiments.

Contribution

It compares the performance of different symplectic schemes on a complex Hamiltonian system, providing insights into their suitability for large-scale lattice simulations.

Findings

01

Certain schemes show better energy conservation.

02

Trade-offs between computational cost and accuracy are identified.

03

Results guide optimal method selection for similar systems.

Abstract

We investigate the performance of various methods of symplectic integration, which are based on two part splitting of the integration operator, for the numerical integration of multidimensional Hamiltonian systems. We implement these schemes to study the behavior of the one-dimensional quartic Klein Gordon disordered lattice with many degrees of freedom (of the order of a few hundreds) and compare their efficiency for the weak chaos regime of the system's dynamics. For this reason we perform extensive simulations for each considered integration scheme. In the process, the second moment and the participation number of the propagating wave packets, along with the system's relative energy error and the required CPU time are registered and compared.

Equations24

H (q, p) = i \sum \frac{p _{i}^{2}}{2} + \frac{ϵ _{i}}{2} q_{i}^{2} + \frac{1}{4} q_{i}^{4} + \frac{1}{2 W} (q_{i + 1} - q_{i})^{2},

H (q, p) = i \sum \frac{p _{i}^{2}}{2} + \frac{ϵ _{i}}{2} q_{i}^{2} + \frac{1}{4} q_{i}^{4} + \frac{1}{2 W} (q_{i + 1} - q_{i})^{2},

\frac{d q _{i}}{d t} = p_{i}, \frac{d p _{i}}{d t} = - ϵ_{i} q_{i} - q_{i}^{3} + \frac{1}{W} (q_{i + 1} + q_{i - 1} - 2 q_{i})

\frac{d q _{i}}{d t} = p_{i}, \frac{d p _{i}}{d t} = - ϵ_{i} q_{i} - q_{i}^{3} + \frac{1}{W} (q_{i + 1} + q_{i - 1} - 2 q_{i})

z (t) = i \geq 0 \sum \frac{t ^{i}}{i !} L_{H}^{i} z_{0} = e^{t L_{H}} z_{0} .

z (t) = i \geq 0 \sum \frac{t ^{i}}{i !} L_{H}^{i} z_{0} = e^{t L_{H}} z_{0} .

A = i \sum \frac{p _{i}^{2}}{2}, B = i \sum \frac{ϵ _{i}}{2} q_{i}^{2} + \frac{1}{4} q_{i}^{4} + \frac{1}{2 W} (q_{i + 1} - q_{i})^{2}

A = i \sum \frac{p _{i}^{2}}{2}, B = i \sum \frac{ϵ _{i}}{2} q_{i}^{2} + \frac{1}{4} q_{i}^{4} + \frac{1}{2 W} (q_{i + 1} - q_{i})^{2}

S A B A_{2} = e^{c_{1} τ L_{A}} e^{d_{1} τ L_{B}} e^{c_{2} τ L_{A}} e^{d_{1} τ L_{B}} e^{c_{1} τ L_{A}}

S A B A_{2} = e^{c_{1} τ L_{A}} e^{d_{1} τ L_{B}} e^{c_{2} τ L_{A}} e^{d_{1} τ L_{B}} e^{c_{1} τ L_{A}}

S B A B_{2} = e^{c_{1} τ L_{B}} e^{d_{1} τ L_{A}} e^{c_{2} τ L_{B}} e^{d_{1} τ L_{A}} e^{c_{1} τ L_{B}}

S B A B_{2} = e^{c_{1} τ L_{B}} e^{d_{1} τ L_{A}} e^{c_{2} τ L_{B}} e^{d_{1} τ L_{A}} e^{c_{1} τ L_{B}}

C = e^{- τ^{3} ε^{2} \frac{c}{2} L_{{{A, B}, B}}}

C = e^{- τ^{3} ε^{2} \frac{c}{2} L_{{{A, B}, B}}}

A B A 82 = e^{a_{1} τ L_{A}} e^{b_{1} τ L_{B}} e^{a_{2} τ L_{A}} e^{b_{2} τ L_{B}} e^{a_{3} τ L_{A}} e^{b_{2} τ L_{B}} e^{a_{2} τ L_{A}} e^{b_{1} τ L_{B}} e^{a_{1} τ L_{A}}

A B A 82 = e^{a_{1} τ L_{A}} e^{b_{1} τ L_{B}} e^{a_{2} τ L_{A}} e^{b_{2} τ L_{B}} e^{a_{3} τ L_{A}} e^{b_{2} τ L_{B}} e^{a_{2} τ L_{A}} e^{b_{1} τ L_{B}} e^{a_{1} τ L_{A}}

S_{2} Y 4 (τ) := S_{2} (a_{1} τ) S_{2} (a_{0} τ) S_{2} (a_{1} τ)

S_{2} Y 4 (τ) := S_{2} (a_{1} τ) S_{2} (a_{0} τ) S_{2} (a_{1} τ)

S z 4 (τ) = L F (k z) L F ((1 - 2 k) z) L F (p z)

S z 4 (τ) = L F (k z) L F ((1 - 2 k) z) L F (p z)

F R o 4 = e^{a_{1} τ L_{A}} e^{b_{1} τ L_{B}} e^{a_{2} τ L_{A}} e^{b_{2} τ L_{B}} e^{a_{3} τ L_{A}} e^{b_{3} τ L_{B}} e^{a_{4} τ L_{A}} e^{b_{3} τ L_{B}} e^{a_{3} τ L_{A}} e^{b_{2} τ L_{B}} e^{a_{2} τ L_{A}} e^{b_{1} τ L_{B}} e^{a_{1} τ L_{A}}

F R o 4 = e^{a_{1} τ L_{A}} e^{b_{1} τ L_{B}} e^{a_{2} τ L_{A}} e^{b_{2} τ L_{B}} e^{a_{3} τ L_{A}} e^{b_{3} τ L_{B}} e^{a_{4} τ L_{A}} e^{b_{3} τ L_{B}} e^{a_{3} τ L_{A}} e^{b_{2} τ L_{B}} e^{a_{2} τ L_{A}} e^{b_{1} τ L_{B}} e^{a_{1} τ L_{A}}

h_{i} = \frac{p _{i}^{2}}{2} + \frac{ϵ _{i}}{2} q_{i}^{2} + \frac{1}{4} q_{i}^{4} + \frac{1}{4 W} (q_{i + 1} - q_{i})^{2}

h_{i} = \frac{p _{i}^{2}}{2} + \frac{ϵ _{i}}{2} q_{i}^{2} + \frac{1}{4} q_{i}^{4} + \frac{1}{4 W} (q_{i + 1} - q_{i})^{2}

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TopicsQuantum chaos and dynamical systems · Nonlinear Photonic Systems · Nonlinear Dynamics and Pattern Formation

Full text

On the symplectic integration of the Klein

Gordon lattice model

B. Senyange

Department of Mathematics and Applied Mathematics

University of Cape Town, Rondebosch, 7701, South Africa

Abstract.

We investigate the performance of various methods of symplectic integration, which are based on two part splitting of the integration operator, for the numerical integration of multidimensional Hamiltonian systems. We implement these schemes to study the behaviour of the one-dimensional quartic Klein Gordon disordered lattice with many degrees of freedom (of the order of a few hundreds) and compare their efficiency for the weak chaos regime of the system’s dynamics. For this reason we perform extensive numerical simulations for each considered integration scheme. In this process, the second moment and the participation number of propagating wave packets, along with the system’s relative energy error and the required CPU time are registered and compared.

Keywords: symplectic integration, Klein-Gordon lattice, disordered systems, Hamiltonian systems

1.

**Introduction

**Symplectic integrators (SIs) are known to preserve the symplectic nature of the Hamiltonian system and keep bounded the error of the computed value of the Hamiltonian. This is one of the advantages that these integrators have over general purpose integrators. In ([2],[21],[4]) various SIs have been applied in the study of the chaotic behavior of two one-dimension Hamiltonian lattices, namely the Klein-Gordon (KG) chain and the Discrete NonLinear Schr $\ddot{o}$ dinger model. These studies showed that there exist different dynamical behaviors, namely the so called weak chaos, strong chaos and the self trapping regime. In this study we consider a wider range of SIs for integration of multidimensional Hamiltonian systems.

In the next section we give a brief discussion of the KG lattice as the Hamiltonian model to use in this study. We also give a description of SIs of generalised order, order two and order four with an insight of how composition techniques are used to generate schemes of higher order. Section $3$ is devoted to comparing the performance of these SIs for the integration of the KG lattice after which we present our conclusions in section $4$ . 2. 2.

The KG Hamiltonian model and the integration schemes

The Hamiltonian $H$ of the one-dimensional KG lattice model of coupled anharmonic oscillators with $N$ degrees of freeedom is

[TABLE]

where $q_{i}$ and $p_{i}$ are the generalised position and momenta of site $i$ respectively. $\epsilon_{i}$ are potential strengths that are chosen uniformly from the interval $[\frac{1}{2},\frac{3}{2}]$ , and $W$ is a parameter that determines the extent of disorder in the lattice. From (0.1), the resulting equations of motion

[TABLE]

can be written as ${\frac{d{\bf z}}{dt}=\{{\bf z},H\}=:L_{H}{\bf z}}$ , where ${{\bf z}=({\bf q},{\bf p})}$ , $L_{H}$ is the so called Poisson bracket that is defined by ${\{F,G\}:=\sum_{i}\left(\frac{\partial F}{\partial q_{i}}\frac{\partial G}{\partial p_{i}}-\frac{\partial F}{\partial p_{i}}\frac{\partial G}{\partial q_{i}}\right)}$ , for differentiable functions $F({\bf z})$ and $G({\bf z})$ . Using initial conditions ${\bf z_{0}}={\bf z}(0)$ , we therefore get a formal solution

[TABLE]

The Hamiltonian (0.1) can be split into two integrable parts as $H({\bf z})=A({\bf p})+B({\bf q})$ where

[TABLE]

and the action of the operators $e^{\tau L_{A}}$ and $e^{\tau L_{B}}$ is known analytically.

A SI approximates the operator $e^{\tau L_{H}}$ by a product of operators $e^{a_{i}\tau L_{A}}$ and $e^{b_{i}\tau L_{B}}$ where the constants $a_{i}$ and $b_{i}$ are chosen depending on the required order of the SI [14].

In our study we consider the performance of order two, order four and generalised order SIs $ABA82,$ $ABA864,$ $ABAH864$ of [22] in integrating system (0.1).

The order two SIs $SABA_{2}$ and $SBAB_{2}$ [3, 5] are

[TABLE]

where $c_{1}=\frac{1}{2}-\frac{1}{2\sqrt{3}}$ , $c_{2}=\frac{1}{\sqrt{3}}$ , $d_{1}=\frac{1}{2}$ , and

[TABLE]

where $d_{1}=\frac{1}{2}$ , $c_{1}=\frac{1}{6}$ and $c_{2}=\frac{2}{3}$

The order of the SIs $SABA_{2}$ and $SBAB_{2}$ can be improved to order four by including a corrector term

[TABLE]

where the value of $c$ is $\frac{(2-\sqrt{3})}{24}$ for $SABA_{2}$ and $\frac{1}{72}$ for $SBAB_{2}$ . We therefore get the so called $SABA_{2}$ with corrector, $SABA_{2}wc={\bf C}SABA_{2}{\bf C}$ and $SBAB_{2}$ with corrector $SBAB_{2}wc={\bf C}SBAB_{2}{\bf C}$

We also consider SIs of generalised order $(8,2)$ and $(8,6,4)$ [6, 12].

Order $(8,2)$ SI $ABA82$ :

[TABLE]

with the coefficients $a_{i}$ , $b_{i}$ stated in [22] and order $(8,6,4)$ SIs $ABA864$ and $ABAH864$ as defined in [12] together with the corresponding coefficients.

For a symmetric order two integrator $S_{2}$ , an order four integrator $S_{2}Y4$ was constructed in [7] by Yoshida using composition techniques. That is to say,

[TABLE]

where

$a_{0}=\frac{2^{\frac{1}{3}}}{2-2^{\frac{1}{3}}}$ and $a_{1}=\frac{1}{2-2^{\frac{1}{3}}}$ . In particular we study the behavior of order four SIs $SABA_{2}Y4$ and $SBAB_{2}Y4$

In [8], using the order two SI Leap-Frog ( $LF$ ) an order four scheme $Sz4$ was constructed:

[TABLE]

where . It can be easily seen that when the Yoshida composition technique is applied to $LF$ , one ends up with $Sz4$ .

Forest and Ruth constructed a fourth order SI which we shall call $FRo4$ defined as

[TABLE]

with the coefficients $a_{i}$ , $b_{i}$ as specified in [9]. 3. 3.

Numerical results

We consider a disorder realization of $H$ in (0.1) for a total of $1000$ sites with a random value of $\epsilon_{i}$ at site $i$ . Fixing $W=4$ and [math] initial displacement we make an initial excitation of the central site with a total energy of $0.4$ . We then keep track of the second moment $m2$ , participation number $P$ and CPU time.

The energy of site $i$ at a time $t$ is

[TABLE]

and $\bar{i}:=\sum_{i}i\frac{h_{i}}{E_{t}}$ .

With energy $E_{t}$ at time $t$ , a normalised energy distribution $\frac{h_{i}}{E_{t}}$ of site $i$ , $m2=\sum_{i}(i-\bar{i})^{2}\frac{h_{i}}{E_{t}}$ is a measure of the rate at which the wave packet spreads from the initially excited central site to all sites in the lattice and $P=\frac{1}{\sum_{i}\left(\frac{h_{i}}{E_{t}}\right)^{2}}$ quantifies the proportion of excited sites in the entire lattice.

In order to compare the performance of the different SIs, we adjust the time step $\tau$ so that the absolute relative energy error $REe:=\left|\frac{E_{t}-E_{0}}{E_{0}}\right|\lesssim 10^{-5}$ at a time $t$ of the evolution; where $E_{0}$ and $E_{t}$ are the energies of the system at times [math] and $t$ respectively. For each of the SIs we ensure that there is a global consistance amongst the SIs in the evolution of $m2$ and $P$ for capturing the dynamics of the wave packet. For each of the SIs we then record the CPU time which is required to perform the simulations.

FIGURE 1 shows the results obtained when we integrate (0.2) using order two schemes $SBAB_{2}$ (red curve) and $SABA_{2}$ (green curve) and generalised order scheme $ABA82$ (gray curves). From this figure we see that with all the schemes portraying practically the same dynamical behavior of the wave packet with respect to $m2$ and $P$ , $ABA82$ has the best performance since it requires the least CPU time compared to the other SIs.

In FIGURE 2 we have results for the integration of equations of motion 0.1 using order four schemes $SABA2wc$ (red curve), $Sz4$ (blue curve), $SBAB_{2}Y4$ (pink curve) and generalized order schemes $ABAH864$ (green curve) and $ABA82$ (gray curves). The generalised order scheme $ABA82$ requires the highest CPU time whereas the schemes $SABA2wc$ and $ABAH864$ show a better performance compared to other schemes since they require the least CPU time.

In FIGURE 3 we have the results for the integration when we use $SABA_{2}Y4$ (red curves), $SBAB_{2}wc$ (green curve), $ABAH864$ (gray curve), $ABA864$ (pink curve) and $FRo4$ (light blue curves). The SIs of generalised order $ABAH864$ and $ABA864$ show a better performance compared to all the other SIs that have been used in the simulations. The SI $ABA864$ reveals the best performance in terms of least CPU time amongst all the SIs that were used in this work.

4.

**Summary and conclusions

**In this work we have studied the integration of the Klein-Gordon lattice model for the so called weak chaos regime. We have used SIs of order two, four and or generalised order. The class of schemes of generalised order have proven to perform better compared to the other schemes that have been tested in this study. Of the three schemes of generalised order, $ABA864$ performed better than $ABAH864$ and $ABA82$ in the integration of the KG model.

Acknowledgments

I would like to thank Muni University for supporting his PhD work at the University of Cape Town through the ADBV HEST project and for facilitating him to attend the EAUMP conference at Makerere University. I am grateful for the input from my supervisor Dr. Ch. Skokos.

Bibliography22

The reference list from the paper itself. Each links out to its DOI / PubMed record.

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