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 Open Access
Some robust integrators for large time dynamics
 Dina Razafindralandy^{1}Email authorView ORCID ID profile,
 Vladimir Salnikov^{2},
 Aziz Hamdouni^{1} and
 Ahmad Deeb^{1}
https://doi.org/10.1186/s4032301901302
© The Author(s) 2019
 Received: 27 November 2018
 Accepted: 9 March 2019
 Published: 28 March 2019
Abstract
This article reviews some integrators particularly suitable for the numerical resolution of differential equations on a large time interval. Symplectic integrators are presented. Their stability on exponentially large time is shown through numerical examples. Next, Dirac integrators for constrained systems are exposed. An application on chaotic dynamics is presented. Lastly, for systems having no exploitable geometric structure, the Borel–Laplace integrator is presented. Numerical experiments on Hamiltonian and nonHamiltonian systems are carried out, as well as on a partial differential equation.
Keywords
 Symplectic integrators
 Dirac integrators
 Longtime stability
 Borel summation
 Divergent series
Introduction
In many domains of mechanics, simulations over a large time interval are crucial. This is, for instance, the case in molecular dynamics, in weather forecast or in astronomy. While many time integrators are available in literature, only few of them are suitable for large time simulations. Indeed, many numerical schemes fail to correctly predict the expected physical phenomena such as energy preservation, as the simulation time grows.
For equations having an underlying geometric structure (Hamiltonian systems, variational problems, Lie symmetry group, Dirac structure, \(\dots \)), geometric integrators appear to be very robust for large time simulation. These integrators mimic the geometric structure of the equation at the discrete scale.
The aim of this paper is to make a review of some time integrators which are suitable for large time simulations. We consider not only equations having a geometric structure but also more general equations. We first present symplectic integrators for Hamiltonian systems. We show in “Symplectic integrators” section their ability in preserving the Hamiltonian function and some other integrals of motion. Applications will be on a periodic Toda lattice and on nbody problems. To simplify, the presentation is done in canonical coordinates.
In “Dirac integrators” section, we show how to fit a constrained problem into a Dirac structure. We then detail how to construct a geometric integrator respecting the Dirac structure. The presentation will be simplified, and the (although very interesting) theoretical geometry is skipped. References will be given for more indepth understanding. A numerical experiment, showing the good long time behaviour of Dirac integrators, will be carried out.
In “Borel–Laplace integrator” section, we present the Borel–Padé–Laplace integrator (BPL). BPL is a generalpurpose time integrator, based on a time series decomposition of the solution, followed by a resummation to enlarge the validity of the series and then reducing the numerical cost on a large time simulation. Finally, the long time behaviour will be investigated through numerical experiments on Hamiltonian and nonHamiltonian system, as well as on a partial differential equation. Numerical cost will be examined when relevant.
Symplectic integrators
We first make some reminder on Hamiltonian systems and their flows in canonical coordinates. Some examples of symplectic integrators are given afterwards and numerical experiments are presented.
Hamiltonian system
In the sequel, H is assumed autonomous in time. It can then be shown that H is preserved along trajectories.
Flow of a numerical scheme
Some symplectic integrators
In the next subsection, some interesting numerical properties of symplectic schemes are highlighted through some model problems.
Numerical experiments
Periodic Toda lattice
To analyze the robustness of the schemes, \({\Delta }t\) is now set to \(10^{1}\). With this time step, the error of the classical Runge–Kutta scheme increases quickly from the first iterations, as can be observed in Fig. 3. It reaches 50% around \(t=4.2 \cdot 10^{3}\). As for it, the RK4sym error oscillates around \(2.26 \cdot 10^{3}\) but does not present any increasing global tendency. Its highest value is about \(3.27 \cdot 10^{3}\), as can be checked in the figure.
It is clear from these experiments that symplectic schemes are particularly stable for large time simulations, where the user wishes a time step as large as possible to reduce the computation cost. In some situation, even a symplectic scheme with a smaller order gives better results over a long time than a classical integrator.
nbody problem
These numerical experiments show again that symplectic schemes are more robust than classical ones for long time dynamics simulation. They have a good behaviour regarding the preservation of the Hamiltonian and some other first integrals despite the perturbation introduced in the initial configuration. Similar results have been obtained in a previous work [24] on an harmonic oscillator, on Kepler’s problem and on vortex dynamics.
Obviously, not all mechanical systems fit into the Hamiltonian formalism, hence there are other geometric constructions that are worth being considered in the context of structurepreserving integrators.
Dirac integrators
In this section, we give an overview of the socalled Dirac structures and describe a class of mechanical systems where those appear naturally, namely systems with constraints. Originally, Dirac structures appear in the work of Courant [6]. The initial motivation was coming from mechanics. As is known, for mechanical systems one can choose between Lagrangian and Hamiltonian formalisms, both being equivalent in finite dimension. The rough idea behind Dirac structures is to consider both formalisms simultaneously, i.e. working with velocities and momenta, however not forgetting that those are dependent variables. Geometrically, this means that instead of choosing between the tangent and cotangent bundles TM or \(T^*M\) for the phase space, we consider their direct sum \(E = TM \oplus T^*M\) and a subbundle of it, subject to some compatibility conditions. Somehow, the original work did not have direct applications to mechanics, since the geometry of the problem turned out to be rather intricate, and gave rise to a lot of development in higher structures and in theoretical physics. In the last decade, however, it was revived with the introduction of socalled port–Hamiltonian [32] and implicit Lagrangian systems [33, 34].
Geometric construction
To transfer this construction from \({\mathbb {R}}^d \times V\) to \({\mathbb {R}}^d \times V^*\), one needs to consider double vector bundles [31]. In our simplified setting this means that the space of interest is \({{\mathcal {V}}} = {\mathbb {R}}^{4d}\), where each component has some geometric interpretation. Namely, we consider \({{\mathcal {V}}}\) as the tangent to \({\mathbb {R}}^d \times V^*\). Naturally, \({\mathbb {R}}^d \times V\) is embedded in \({{\mathcal {V}}}\) (recall that V is tangent to \({\mathbb {R}}^d\)). The constraint set is then a subset \({\tilde{\Delta }} \subset {{\mathcal {V}}}\), and the differential forms \(\alpha ^a(\mathbf {q})\) generate its annihilator \(\Delta _0\) that naturally belongs to \({{\mathcal {V}}}^*\). Note that, since \({\mathbb {R}}^d \times V^*\) is a symplectic space, it is equipped with a bilinear antisymmetric nondegenerate closed form \(\Omega \) (this form \(\Omega \) is the generalisation of the matrix \(\mathbb {J}\) of “Symplectic integrators” section in noncanonical coordinates). One can then construct a symplectic mapping \(\Omega ^\flat :{{\mathcal {V}}} \rightarrow {{\mathcal {V}}}^*\).
Discretization
It is important to note that the previous section is not just a “fancy” way of recovering the wellknown theory: every step of the construction admits a discrete analog. We briefly present the recipe of this discretization and, again, refer the interested reader to [27] for details and examples.

\(\mathbf {v}^n := \frac{\mathbf {q}^{n+1}  \mathbf {q}^n}{{\Delta }t}\), we label it Dirac1, and

\(\mathbf {v}^n := \frac{\mathbf {q}^{n+1}  \mathbf {q}^{n1}}{2{\Delta }t}\), labelled Dirac2.
It is important to note that, in some sense, this construction generalizes the previous section. If one considers the system without constraints but still applies the procedure, (22) becomes obsolete, and in (21) the righthandsides vanish, so one obtains a numerical method for the dynamics of a Lagrangian system governed by L. By a straightforward computation, one checks that for a natural mechanical system with a potential U, i.e. when \(L = \frac{1}{2}m\mathbf {v}^2 + U(\mathbf {q})\), Dirac1 is symplectic. And it is also meaningful to consider a symplectic version of Dirac2 (we do not detail it here since we would need to explain what is symplecticity for a multistep method).
Example: chaos for double pendulum
From the point of view of the previous subsection, this is a typical example of a system with constraints: the distance \(\ell _1\) from the first mass point to the origin and the distance \(\ell _2\) between the two mass points are fixed. The system admits a parametrization in terms of angles, but we will pretend not to know it, to test the method.
The typical result of simulations is shown in Fig. 10. Dirac2 and explicit Euler methods are compared. For visualization (but not for computation), we use the angle representation of the double pendulum (see Fig. 9). Both algorithms start with the same initial data, and the same timestep \(\Delta t = 0.0001\). They are in good agreement in the beginning as can be seen on the two topgraphics of Fig. 10. But already at time \(T = 50\), the difference is visible (graphics in the middle). And towards \(T=100\) the difference becomes dramatic: for the Euler method the pendulum is making full turns instead of oscillation. And this is clearly a computation artifact, since decreasing the timestep one gets rid of the discrepancy and recovers the left picture for both methods. Note also that Dirac structure based method preserves the constraints much better than the Euler one: the error is \(2.2\cdot 10^{6}\) compared to 0.06 respectively.
A similar effect is observed for other methods: trapezium, and even Runge–Kutta, which is of higher order. Moreover, there is another nonnegligible convenience of the Dirac structure based methods: the Lagrange multipliers are treated like other dynamical variables, there is no need to resolve “by hand” the equation related to constraints (22).
In many areas of mechanics, systems are often described by an underlying geometric structure. As observed in the two previous sections, making use of these structures leads to more robust numerical schemes. In the last section, we propose an integrator which is suitable to general systems where no geometric structure is exploitable for numerical simulations.
Borel–Laplace integrator

Given an initial condition \(u(t_0)=u_0\), compute a truncated series solution via recurrence (26): \(\displaystyle \breve{u}^N(t)=\sum \nolimits _{n=0}^{N}u_nt^n \) .

Compute its Borel transform: \(\displaystyle \mathcal {B}\breve{u}^N({\xi })=\sum \nolimits _{n=0}^{N1}\frac{u_{n+1}}{n!}\,{\xi }^n.\)

Transform \(\mathcal {B}\breve{\mathbf {u}}^N({\xi })\) into a rational fraction function via a Padé approximation: \(\displaystyle P^N({\xi })=\frac{a_0+a_1t+\dots a_{N_{num}}t^{N_{num}}}{b_0+b_1t+\dots b_{N_{den}}t^{N_{den}}}\)
The Padé approximation materializes the prolongation in the Borel summation procedure.

Apply a Laplace transformation (at 1/t) on \(P({\xi })\) to obtain a numerical Borel sum \(\displaystyle \mathcal {S}\breve{u}^N(t)=\int _0^{+\infty }P^N(\xi )\text {e}^{\xi /t}{\mathrm {d}}\xi .\)
Numerically, the integral is replaced by a Gauss–Laguerre quadrature.

Take \(\mathcal {S}\breve{u}^N(t)\) as an approximate solution u(t) of (23) within the integral \([t_0,t_1]\) where the residue of the equation is smaller than a parameter \({\epsilon }_{res}\).

Restart the algorithm with \(u_0=u(t_1)\) as initial condition to obtain an approximate solution for larger values of t.
An advantage of BPL is that it is totally explicit, in contrast with symplectic integrators in general. Moreover, changing the order of the scheme is as easy as setting N to a different value. Note also that the resummation procedure can be done componentwise, enabling an easy parallelization on multicore computers. However, no such optimization has been done in the present article.
In the following subsection, a partial analysis of the symplecticity property of BPL is presented.
Highorder symplecticity
Lemma 1
This can be straightforwardly deduced by injecting the time series \(\breve{u}\) in (2) and identifying the coefficients of each \(t^n\). Next, if the series is convergent then, inside the convergence disc, \(\breve{u}\) is the exact solution. In this case, \(\breve{{\varphi }}_t\) is symplectic. We reformulate this statement in the following theorem.
Theorem 2
Corollary 3
Theorem 4
In the following subsections, BPL is implemented and tested on a Hamiltonian equation. Next, we present some experiments on nonHamiltonian equations.
In simulations, the truncation order of the series is set to \(N=10\) unless otherwise stated. The degree of the numerator and the denominator of the Padé approximant are \(N_{num}=4\) and \(N_{den}=5\). A singular value decomposition is used to strengthen the robustness of the Padé calculation [11]. Twenty Gauss–Laguerre roots are used for the quadrature.
The aim of these simulations is not to make an extensive comparison of BPL with classical schemes (this will be done in a forthcoming paper) but only to show the potential of the scheme in predicting long time dynamics.
Periodic Toda lattice
We consider again the periodic Toda lattice from “Periodic Toda lattice” section. The quality parameter \({\epsilon }_{res}\) of BPL is choosen such that the mean time step \({\delta }t\) is approximately 0.1, and compare the results with that of RK4 and RK4sym (see Figs. 3 and 4).
Toda lattice
Mean timestep  CPU  Mean error  

RK4  0.1  107.44  \(3.7902.10^{1}\) 
RK4sym  0.1  128.74  \(2.261 \cdot 10^{3}\) 
BPL  0.0983  259.31  \(3.010 \cdot 10^{4}\) 
Toda lattice
Mean timestep  CPU  Mean error  

RK4  0.0275  1475.49  \(2.130 \cdot 10^{3}\) 
RK4sym  0.1  128.74  \(2.261 \cdot 10^{3}\) 
BPL  0.125  179.08  \(2.897 \cdot 10^{3}\) 
Duffing equation

Case 1: \(a=2/9,\,b=1,\,r=1\),

Case 2: \(a=1,\,b=1,\,r=0\).
In the last subsection, BPL is applied to a semidiscretized partial differential equation. It is compared to some other adaptative schemes. Since the system is big enough, it is worth to give an indication on the CPU simulation time.
KortevegdeVries equation
BPL is compared to two other schemes. The first one is the adaptative 4th order Runge–Kutta scheme (still denoted RK4 in this subsection). This scheme is explicit. The second one is the exponential time differencing associated to RK4 (denoted ETDRK4), developed by Cox and Matthews in [7]. This scheme is based on an exact, exponential type, resolution of the linear part of the equation, followed by an explicit adaptative RungeKutta resolution of the nonlinear part. The algorithm is not completely explicit since it requires the (pseudo)inversion of a matrix. Moreover, it generally needs the evaluation of a matrix exponential, which is numerically expensive. This evaluation is done via a Padé approximants in simulations.
Time step, error and CPU time over one period, with \(d=128\)
BPL  ETDRK4  RK4  

Mean time step  \(1.53\,\cdot 10^{01}\)  \(6.41\,\cdot 10^{04}\)  \(1.88\,\cdot 10^{03}\) 
\(L^2\) error at \(t=T\)  \(9.76\,\cdot 10^{06}\)  \(1.10\,\cdot 10^{05}\)  \(1.69\,\cdot 10^{05}\) 
Simulation time  1.74  \(1.66\,\cdot 10^{+03}\)  \(5.98\,\cdot 10^{+01}\) 
In the next simulation, we analyse the behaviour of the schemes when the size d of the problem is increased. Figure 17a presents the \(L^2\) error for \(t=T\). It shows that the precision of BPL and ETDRK4 remains approximately the same, except when d is very small. Figure 17b shows however that BPL requires much less iterations (100 iterations versus 20389 for ETDRK4 when \(d=512\) to reach one period). The time steps of BPL and ETDRK4 seem to have the same behaviour when d is large enough. Indeed, they tend to be independent of d, as suggested by Fig. 17b. However, the computation time increases much more rapidly with ETDRK4. For BPL, the growth rate of the CPU time between \(d=128\) and \(d=512\) is 51 percent whereas, for ETDRK4, it is 381 percent.
In all of the previous simulations, the truncation order N of the time series in BPL was set to 10. In our last test, the effect of N on the quality of BPL is analysed. For this, the size of the problem is fixed to \(d=128\). Figure 18a shows that the time step increases with N, passing from \({\Delta }t_{mean}=0.0256\) for \(N=4\) to \({\Delta }t_{mean}=0.156\) when \(N=14\). Despite the number of iterations is consequently reduced, the CPU time also increases with N, going from 0.686 to 0.173 s, as can be observed in Fig. 18b. This is caused by the fact that more coefficients of the series and more Padé coefficients have to be computed. As for it, the error fluctuates but globally decreases from \(7.51\cdot 10^{5}\) to \(4.30\cdot 10^{6}\). This fluctuation is not uncommon in series based approximations. It is interesting to note that whereas the error is divided by 17.5, the CPU time is multiplied only by 3.96 between \(N=4\) and \(N=14\). In other words, the precision increases faster than the CPU time when the order of the scheme is increased.
Conclusion
In this article, we gave an overview of some time integrators for longtime simulations. Two geometric integrators and a generalpurpose time integrator was presented.
Through numerical examples, the ability of symplectic integrator in preserving the Hamiltonian, the angular momentum or eigenvalues was observed. Moreover, it was shown that symplectic integrators are more robust compared to classical schemes when the time step is enlarged (in the example of Toda lattice) or when a perturbation is introduced (threebody problem).
Next, a way of constructing Dirac integrators for constrained system was given. Numerical experiments showed that respecting the Dirac structure at discrete level avoids numerical artifacts. As a consequence, Dirac integrators are able to reproduce the dynamics of the system over a long time.
Finally, we showed that BPL competes with symplectic integrators in predicting Hamiltonian dynamics (Toda lattice and Case 2 of Duffing equation). For more general equations, BPL also preserves with high precision the first integral of the system, as well as the periodicity when the solution is periodic. Lastly, compared to two popular schemes, BPL appears to require less computation time.
Declarations
Author's contributions
All the authors contributed and participated to the elaboration of the article. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
Please contact author for data requests.
Funding
Not applicable.
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