Symplectic Runge-Kutta Methods Research Articles

Linearly Implicit Conservative Schemes for the Nonlocal Schrödinger Equation

This paper introduces two high-accuracy linearly implicit conservative schemes for solving the nonlocal Schrödinger equation, employing the extrapolation technique. These schemes are based on the generalized scalar auxiliary variable approach and the symplectic Runge–Kutta method. By integrating these advanced methods, the proposed schemes aim to significantly enhance computational accuracy and efficiency, while maintaining the essential conservative properties necessary for accurate physical modeling. This offers a structured approach to handle auxiliary variables, ensuring stability and conservation, while the symplectic Runge–Kutta method provides a robust framework with high accuracy. Together, these techniques offer a powerful and reliable approach for researchers dealing with complex quantum mechanical systems described by the nonlocal Schrödinger equation, ensuring both accuracy and stability in their numerical simulations.

Diagonally Implicit Symplectic Runge-Kutta Methods with 7th Algebraic Order

The numerical integration of Hamiltonian systems with oscillating solutions is considered in this paper. Since Hamiltonian systems have good properties such as symplecticity, numerical methods that preserve these properties have attracted the great attention. In fact, the explicit Runge-Kutta methods have used due to that schemes are very simple and its computational amounts are very small. However, the explicit schemes aren’t stable so the implicit Runge-Kutta methods have widely studied. Among those implicit schemes, symplectic numerical methods were interested. It is because it has preserved the original property of the systems. So, study of the symplectic Runge-Kutta methods have performed. The typical symplectic Runge-Kutta method is the Gauss-Legendre method, whose drawback is that it is a general implicit scheme and is too computationally expensive. Despite these drawbacks, the study of the diagonally implicit symplectic Runge-Kutta methods that preserves symplecticity has attracted much attention. The symplectic Runge-Kutta method has been studied up to sixth order in the past and efforts to obtain higher order conditions and algorithms are being intensified. In many applications such as molecular dynamics as well as in space science, such as satellite relative motion studies, this method is very effective and its application is wider. In this paper, it is presented the 7<sup>th</sup> order condition and derive the corresponding optimized method. So the diagonally implicit symplectic eleven-stages Runge-Kutta method with algebraic order 7 and dispersion order 8 is presented. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing same type Runge-Kutta methods.

Novel structure-preserving schemes for stochastic Klein–Gordon–Schrödinger equations with additive noise

Stochastic Klein–Gordon–Schrödinger (KGS) equations are important mathematical models and describe the interaction between scalar nucleons and neutral scalar mesons in the stochastic environment. In this paper, we propose novel structure-preserving schemes to numerically solve stochastic KGS equations with additive noise, which preserve averaged charge evolution law, averaged energy evolution law, symplecticity, and multi-symplecticity. By applying central difference, sine pseudo-spectral method, or finite element method in space and modifying finite difference in time, we present some charge and energy preserved fully-discrete schemes for the original system. In addition, combining the symplectic Runge-Kutta method in time and finite difference in space, we propose a class of multi-symplectic discretizations preserving the geometric structure of the stochastic KGS equation. Finally, numerical experiments confirm theoretical findings.

Dynamic behaviors of hierarchical-tethered towing system for space debris removal

The debris-tether-tug (DTT) system, has been considered as one of the most promising techniques for the space debris removal. In order to decrease the control difficulty of the DTT system, the DTT system was usually designed with a single tether. However, when the target owns the complex attitude motion, the removal efficiency of the classic DTT system (towing the target by a single tether) is limited. A modified DTT model with the hierarchical-tethered architecture is proposed to improve the removal efficiency in this paper. Based on the Hamiltonian variational principle, the coupling non-smooth dynamic model for the hierarchical-tethered towing system is established, in which, four tethers are divided into two groups to describe the asymmetry of the DTT model well. The symplectic Runge-Kutta method is employed to simulate the evolution of the orbit radius of the target, the length of the tethers, the position of the tethers’ mass center and the target attitude angle. The numerical results reported show that, the competition relationship between the attitude stability of the target and the stability of the variation of the sub-tethers is affected by the relative stiffness of the tethers. In addition, the capture ability of the modified DTT system for the space target with a certain spinning speed is illustrated, which provides a theoretical basis for the design of the DTT system in future.

High-order Runge–Kutta structure-preserving methods for the coupled nonlinear Schrödinger–KdV equations

A novel class of high-order Runge–Kutta structure-preserving methods for the coupled nonlinear Schrödinger–KdV equations is proposed and analyzed. With the aid of the quadratic auxiliary variable, an equivalent system is obtained from the original problem. The Fourier pseudo-spectral method is employed in spatial discretization and the symplectic Runge–Kutta method is utilized for the resulting semi-discrete system to arrive at a high-order fully discrete scheme. Simultaneously, the conservation of the original multiple invariants for the schemes are rigorously proven. Numerical experiments are performed to verify the theoretical analysis.

High-order schemes for the fractional coupled nonlinear Schrödinger equation

<abstract><p>This paper considers the fractional coupled nonlinear Schrödinger equation with high degree polynomials in the energy functional that cannot be handled by using the quadratic auxiliary variable method. To this end, we develop the multiple quadratic auxiliary variable approach and then construct a family of structure-preserving schemes with the help of the symplectic Runge-Kutta method for solving the equation. The given schemes have high accuracy in time and can both inherit the mass and Hamiltonian energy of the system. Ample numerical results are given to confirm the accuracy and conservation of the developed schemes at last.</p></abstract>

Arbitrary high-order linearly implicit energy-conserving schemes for the Rosenau-type equation

In this paper, we present a class of high-order linearly implicit energy-conserving schemes for solving the generalized Rosenau-type equation. We firstly reformulate the equation into a skew-adjoint system with the energy conservation law. Then the nonlinear terms of the system are approximated with an extrapolation/prediction–correction technique and then a linearized energy-conserving system is obtained. Finally, the symplectic Runge–Kutta method in time and the Fourier pseudo-spectral method in space are employed to discretize the resulting linearized system and a fully discrete scheme is obtained, which is linearly implicit, high-order and energy-conserving. Numerical results are addressed to demonstrate the accuracy and efficiency of the proposed scheme.

Arbitrary high-order exponential integrators conservative schemes for the nonlinear Gross-Pitaevskii equation

In this paper, we propose a family of high-order conservative schemes based on the exponential integrators technique and the symplectic Runge-Kutta method for solving the nonlinear Gross-Pitaevskii equation. By introducing generalized scalar auxiliary variable, the equation is equivalent to a new system with both mass and modified energy conservation laws. Then, a conservative semi-discrete exponential scheme is given by combining the symplectic Runge-Kutta method and the Lawson method in the time direction. Subsequently, we apply the Fourier pseudo-spectral method to approximate semi-discrete system in space and obtain the fully-discrete schemes that conserve the energy and mass. Numerical examples are presented to confirm the accuracy and conservation of the developed schemes.

How do Monte Carlo estimates affect stochastic geometric numerical integration?

In this work, we investigate the numerical conservation of characteristic properties of stochastic Hamiltonian problems driven by additive noise under time discretizations, in the spirit of stochastic geometric numerical integration. In particular, we aim to understand how Monte Carlo and multilevel Monte Carlo estimators for the expected values eventually affect the conservation properties of the numerical scheme. Specifically, our analysis is focused on the role of random number generation in the conservation of invariant laws for stochastic Hamiltonian problems under time discretization by the drift-preserving numerical scheme introduced in C. Chen, D. Cohen, R. D'Ambrosio and A. Lang, [Drift-preserving numerical integrators for stochastic Hamiltonian systems, Adv. Comput. Math. 46(2) (2020), pp. 27.] and the stochastic perturbation of a symplectic Runge–Kutta method, introduced in K. Burrage and P.M. Burrage, [Low rank Runge–Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise, J. Comput. Appl. Math. 236(16) (2012), pp. 3920–3930.]. The numerical evidence confirms the aforementioned theoretical analysis.

Variational symplectic diagonally implicit Runge-Kutta methods for isospectral systems

Isospectral flows appear in a variety of applications, e.g. the Toda lattice in solid state physics or in discrete models for two-dimensional hydrodynamics. The isospectral property thereby often corresponds to mathematically or physically important conservation laws. The most prominent feature of these systems, i.e. the conservation of the eigenvalues of the matrix state variable, should therefore be retained when discretizing these systems. Recently, it was shown how isospectral Runge-Kutta methods can in the Lie-Poisson case be obtained through Hamiltonian reduction of symplectic Runge-Kutta methods on the cotangent bundle of a Lie group. We provide the Lagrangian analogue and, in the case of symplectic diagonally implicit Runge-Kutta methods, derive the methods through a discrete Euler-Poincaré reduction. Our derivation relies on a formulation of diagonally implicit isospectral Runge-Kutta methods in terms of the Cayley transform, generalizing earlier work that showed this for the implicit midpoint rule. Our work is also a generalization of earlier variational Lie group integrators that, interestingly, appear when these are interpreted as update equations for intermediate time points. From a practical point of view, our results allow for a simple implementation of higher order isospectral methods and we demonstrate this with numerical experiments where both the isospectral property and energy are conserved to high accuracy.

A Novel Class of Energy-Preserving Runge-Kutta Methods for the Korteweg-de Vries Equation

In this paper, we present a quadratic auxiliary variable approach to develop a new class of energy-preserving Runge-Kutta methods for the Korteweg-de Vries equation. The quadratic auxiliary variable approach is first proposed to reformulate the original model into an equivalent system, which transforms the energy conservation law of the Korteweg-de Vries equation into two quadratic invariants of the reformulated system. Then the symplectic Runge-Kutta methods are directly employed for the reformulated model to arrive at a new kind of time semi-discrete schemes for the original problem. Under the consistent initial condition, the proposed methods are rigorously proved to maintain the original energy conservation law of the Korteweg-de Vries equation. In addition, the Fourier pseudo-spectral method is used for spatial discretization, resulting in fully discrete energy-preserving schemes. To implement the proposed methods effectively, we present a very efficient iterative technique, which not only greatly saves the calculation cost, but also achieves the purpose of practically preserving structure. Ample numerical results are addressed to confirm the expected order of accuracy, conservative property and efficiency of the proposed algorithms.

Arbitrary high-order linear structure-preserving schemes for the regularized long-wave equation

In this paper, a class of arbitrarily high-order linear momentum-preserving and energy-preserving schemes are proposed, respectively, for solving the regularized long-wave equation. For the momentum-preserving scheme, the key idea is based on the extrapolation/prediction-correction technique and the symplectic Runge-Kutta method in time, together with the standard Fourier pseudo-spectral method in space. We show that the scheme is linear, high-order, unconditionally stable and preserves the discrete momentum of the system. For the energy-preserving scheme, it is mainly based on the energy quadratization approach and the analogous linearized strategy used in the construction of the linear momentum-preserving scheme. The proposed scheme is linear, high-order and can preserve both the discrete mass and the discrete quadratic energy exactly. Numerical results are addressed to demonstrate the accuracy and efficiency of the proposed scheme.

Arbitrary high-order linearly implicit energy-preserving algorithms for Hamiltonian PDEs

In this paper, we present a novel strategy to systematically construct linearly implicit energy-preserving schemes with arbitrary order of accuracy for Hamiltonian PDEs. Such a novel strategy is based on the newly developed exponential scalar auxiliary variable (ESAV) approach that can remove the bounded-from-below restriction of nonlinear terms in the Hamiltonian functional and provides a totally explicit discretization of the auxiliary variable without computing extra inner products. So it is more effective and applicable than the traditional scalar auxiliary variable (SAV) approach. To achieve arbitrary high-order accuracy and energy preservation, we utilize symplectic Runge-Kutta methods for both solution variables and the auxiliary variable, where the values of the internal stages in nonlinear terms are explicitly derived via an extrapolation from numerical solutions already obtained in the preceding calculation. A prediction-correction strategy is proposed to further improve the accuracy. Fourier pseudo-spectral method is then employed to obtain fully discrete schemes. Compared with the SAV schemes, the solution variables and the auxiliary variable in these ESAV schemes are now decoupled. Moreover, when the linear terms have constant coefficients, the solution variables can be explicitly solved by using the fast Fourier transform. Numerical experiments are carried out for three Hamiltonian PDEs to demonstrate the efficiency and conservation of the ESAV schemes.

Clebsch canonization of Lie–Poisson systems

<p style='text-indent:20px;'>We propose a systematic procedure called the Clebsch canonization for obtaining a canonical Hamiltonian system that is related to a given Lie–Poisson equation via a momentum map. We describe both coordinate and geometric versions of the procedure, the latter apparently for the first time. We also find another momentum map so that the pair of momentum maps constitute a dual pair under a certain condition. The dual pair gives a concrete realization of what is commonly referred to as collectivization of Lie–Poisson systems. It also implies that solving the canonized system by symplectic Runge–Kutta methods yields so-called collective Lie–Poisson integrators that preserve the coadjoint orbits and hence the Casimirs exactly. We give a couple of examples, including the Kida vortex and the heavy top on a movable base with controls, which are Lie–Poisson systems on <inline-formula><tex-math id="M1">\begin{document}$ \mathfrak{so}(2,1)^{*} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ (\mathfrak{se}(3) \ltimes \mathbb{R}^{3})^{*} $\end{document}</tex-math></inline-formula>, respectively.</p>

Symplectic analysis on dynamic behaviors of tethered tug–debris system

As one of the potential active debris removal approaches, removal of space debris by a tether tow may be applied to perform the large space debris removal mission. To remedy the omissions of the current references, including effects of the initial state and the considerable elongation of the tether on the evolution of the tumbling angle of the target debris, the dynamic behaviors of the tethered tug–debris system are investigated by using the symplectic Runge-Kutta method in this paper. Introducing the generalized momentum vector for the generalized coordinate vector, the finite-dimensional generalized Hamiltonian function is yielded and the generalized Hamiltonian dual equation describing the coupling dynamic behaviors of the tethered tug–debris system is deduced. To investigate the coupling dynamic behaviors of the tethered tug–debris system, the second-level fourth-order symplectic Runge-Kutta scheme is constructed and the perfect structure-preserving properties of it is verified. The effects of the elastic constant of the tether and the initial Euler angular velocity on the evolution of the tumbling angle are investigated in the numerical experiments in detail. The beat characteristic and the “window” characteristic in the evolution of the tumbling angle with small elastic constants of the tether are reproduced. In addition, the sharp change phenomenon in the evolution of the tumbling angle is found when the elastic constant of the tether is extremely small. The above findings can be used to guide the strategy design of the removal of space debris by a tether tow.

Arbitrarily high order structure-preserving algorithms for the Allen-Cahn model with a nonlocal constraint

We develop fully discrete structure-preserving numerical algorithms of arbitrarily high order for the Allen-Cahn model with a nonlocal constraint subject to the Neumann boundary condition. Using the energy quadratization methodology, we reformulate the thermodynamically consistent model into an equivalent one with a quadratic free energy. For the reformulated model, we first apply a Cosine pseudo-spectral approximation in space to arrive at a semi-discrete system that inherits the volume conservation and energy dissipative property; then we use two distinct temporal discretization methods to derive fully discrete schemes of arbitrarily higher order. One is based on the symplectic Runge-Kutta (RK) method and the other is a linearized Runge-Kutta method by the prediction-correction strategy. The fully discrete schemes preserve both volume and the energy dissipative property. In addition, we show that the liner system resulting from the schemes warrants the unique solvability. A fast solver combined with the discrete Cosine transform (DCT) is exploited to implement the high-order scheme efficiently. Extensive numerical examples are presented to show the efficiency and accuracy of the newly proposed methods.

Deep learning as optimal control problems

We briefly review recent work where deep learning neural networks have been interpreted as discretisations of an optimal control problem subject to an ordinary differential equation constraint. We report here new preliminary experiments with implicit symplectic Runge-Kutta methods. In this paper, we discuss ongoing and future research in this area.

Coupling dynamic behaviors of flexible stretching hub-beam system

In the 1st stage deployment of the IKAROS solar sail, the flexible beam with comparable mass relative to the mass of the rotor is stretching actively in the axial direction, which brings new challenge in the dynamic analysis for the 1st stage deployment of the IKAROS solar sail. Considering the active axial stretching of the flexible beam, a simplified coupling dynamic model for the 1st stage deployment of the IKAROS solar sail is proposed based on the non-holonomic Hamilton least-action principle firstly. And then, a structure-preserving approach combining the generalized multi-symplectic method and the symplectic Runge-Kutta method is constructed to simulate the evolution of the transverse vibration of the beam as well as the evolution of the rotation of the hub. Finally, the associated numerical results in three stages are reported. From the numerical results, it can be found that the coupling effects between the deformation of the beam, the active stretching of the beam and the rotation of the hub are reflected in the stretching stage. In this stage, the transverse vibration of the beam is enhanced by the stretching effect of the beam, and the increase of the energy of the beam in this stage is derived from the decrease of the rotational energy of the hub. In addition, the structure-preserving properties and the validity of the numerical results are verified by the tiny relative energy dissipation of the flexible stretching hub-beam system in the stretching stage.

Orbit-attitude-vibration coupled dynamics of tethered solar power satellite

A new orbit-attitude-vibration coupled dynamic model of the tethered solar power satellite (Tethered SPS) is established based on absolute nodal coordinate formulation. The Hamilton’s equation of the system is derived by introducing generalized momentum through Legendre transformation. The correctness of the proposed model is verified by an example. The dynamic characteristics of the Tethered SPS are studied using the symplectic Runge-Kutta method. Simulation results show that the orbital radius and the total energy of the system are well preserved. The attitude of the system is unstable when the mass of the bus system is small. However, the attitude stability is dependent on some other parameters of the system, which requires further studies. It is also found that the average tether force/deformation can be roughly estimated by simplifying the solar panel as a particle. The proposed model can be used to study the orbit-attitude-vibration coupled dynamics and control problems.

Coupled Orbit-Attitude Dynamics of Tethered-SPS

In this paper, a dynamic model for long-term on-orbit operation of the tethered solar power satellite (Tethered-SPS) is established. Because the Tethered-SPS is a super-large and super-flexible structure, the coupling among the orbit, attitude, and structure vibration of the system should be considered in comparison with the traditional spacecraft dynamic models. Based on the absolute nodal coordinate formulation (ANCF), the dynamic equation of the Tethered-SPS is established by Hamiltonian variational principle, and the dual equations of the Hamiltonian system are established by introducing generalized momentum through Legendre transformation. The symplectic Runge–Kutta method is used for numerical simulation, and the validation of the modeling method is verified by a numerical example. The effects of orbital altitude, initial attitude angle, length of solar panel, and orbital eccentricity on the orbit and attitude of the Tethered-SPS are analyzed. The numerical simulation results show that the effect of orbital altitude and length of solar panel on the orbital error of midpoint of beam is small. However, the initial attitude angle has a significant effect on the orbital error of midpoint of solar panel. The effect of the length of solar panel on the attitude angle of the system is not significant, but orbital altitude, orbital eccentricity, and initial attitude angle of the system severely affect the attitude angle of the system. Then, the stability of the system is affected.