Geometric Numerical Integrators Research Articles

A composite method based on delta‐shaped basis functions and Lie group high‐order geometric integrator for solving Kawahara‐type equations

In this paper, we devise a novel method to solve Kawahara‐type equations numerically. In this novel method, for spatial discretization, we use delta‐shaped basis functions and generate differentiation matrices for spatial derivatives of the Kawahara‐type equations. For discretization of temporal variable, we utilize a high‐order geometric numerical integrator based on Lie group methods. For illustration of efficiency of the suggested method, we consider some test problems. We calculate errors and make some comparisons with other results that exist in literature. We also report changes in conservation laws during numerical simulations, and we indicate that the suggested method can preserve the conservation laws pretty good. Outcomes of numerical simulations indicate that the suggested method in this paper is reliable and effective for nonlinear partial differential equations (PDEs).

Conformal structure‐preserving schemes for damped‐driven stochastic nonlinear Schrödinger equation with multiplicative noise

AbstractSince many physical phenomena are often influenced by dispersive medium, energy compensation and random perturbation, exploring the dynamic behaviors of the damped‐driven stochastic system has becoming a hot topic in mathematical physics in recent years. In this paper, inspired by the stochastic conformal structure, we investigate the geometric numerical integrators for the damped‐driven stochastic nonlinear Schrödinger equation with multiplicative noise. To preserve the conformal structures of the system, by using symplectic Euler method, implicit midpoint method and Fourier pseudospectral method, we propose three kinds of stochastic conformal schemes satisfying corresponding discrete stochastic multiconformal‐symplectic conservation laws and discrete global/local charge conservation laws. Numerical experiments illustrate the structure‐preserving properties of the proposed schemes, as well as favorable results over traditional nonconformal schemes, which are consistent with our theoretical analysis.

A Hamiltonian Formulation on Manifolds for Dynamic Modeling of Constrained Mechanisms and Energy-Preserving Integration

Abstract The dynamic modeling of multibody system is crucial in motion simulations, design, and control of mechanisms. This paper proposes a Hamiltonian formulation on manifolds for mechanism modeling which involves three key steps: (1) the local parameterization of regular configuration space; (2) the coordinate formulation of the Legendre transformation; and (3) the derivation of Hamiltonian equations. Geometric numerical integrators can be naturally deployed on the proposed formulation and achieve a long-time energy-preserving integration. Based on parametric symplectic integrators and the chart-splicing technique, a novel energy-preserving scheme is proposed. Simulations on two constrained mechanisms verify our claims.

A Variational Formulation of Accelerated Optimization on Riemannian Manifolds

It was shown recently by Su et al. (2016) that Nesterov's accelerated gradient method for minimizing a smooth convex function $f$ can be thought of as the time discretization of a second-order ODE, and that $f(x(t))$ converges to its optimal value at a rate of $\mathcal{O}(1/t^2)$ along any trajectory $x(t)$ of this ODE. A variational formulation was introduced in Wibisono et al. (2016) which allowed for accelerated convergence at a rate of $\mathcal{O}(1/t^p)$, for arbitrary $p>0$, in normed vector spaces. This framework was exploited in Duruisseaux et al. (2020) to design efficient explicit algorithms for symplectic accelerated optimization. In Alimisis et al. (2020), a second-order ODE was proposed as the continuous-time limit of a Riemannian accelerated algorithm, and it was shown that the objective function $f(x(t))$ converges to its optimal value at a rate of $\mathcal{O}(1/t^2)$ along solutions of this ODE. In this paper, we show that on Riemannian manifolds, the convergence rate of $f(x(t))$ to its optimal value can also be accelerated to an arbitrary convergence rate $\mathcal{O}(1/t^p)$, by considering a family of time-dependent Bregman Lagrangian and Hamiltonian systems on Riemannian manifolds. This generalizes the results of Wibisono et al. (2016) to Riemannian manifolds and also provides a variational framework for accelerated optimization on Riemannian manifolds. An approach based on the time-invariance property of the family of Bregman Lagrangians and Hamiltonians was used to construct very efficient optimization algorithms in Duruisseaux et al. (2020), and we establish a similar time-invariance property in the Riemannian setting. One expects that a geometric numerical integrator that is time-adaptive, symplectic, and Riemannian manifold preserving will yield a class of promising optimization algorithms on manifolds.

A combination of Lie group-based high order geometric integrator and delta-shaped basis functions for solving Korteweg–de Vries (KdV) equation

In this work, we develop a novel method to obtain numerical solution of well-known Korteweg–de Vries (KdV) equation. In the novel method, we generate differentiation matrices for spatial derivatives of the KdV equation by using delta-shaped basis functions (DBFs). For temporal integration we use a high order geometric numerical integrator based on Lie group methods. This paper is a first attempt to combine DBFs and high order geometric numerical integrator for solving such a nonlinear partial differential equation (PDE) which preserves conservation laws. To demonstrate the performance of the proposed method we consider five test problems. We reckon [Formula: see text], [Formula: see text] and root mean square (RMS) errors and compare them with other results available in the literature. Besides the errors, we also monitor conservation laws of the KDV equation and we show that the method in this paper produces accurate results and preserves the conservation laws quite good. Numerical outcomes show that the present novel method is efficient and reliable for PDEs.

Inclusion of Non-Conservative Forces in Geometric Integrators with Application to Orbit–Attitude Coupling

In this paper, a discretization method for incorporating nonconservative forces in a class of geometric numerical integrators known as variational integrators and Galerkin variational propagators is proposed. The proposed method does not require modification of the original integration algorithm used in conservative systems. First, the damped harmonic oscillator is used as benchmark for evaluating the proposed approach. Two more complex scenarios are presented next: one considering propagations in the two-body problem with drag forces, and another dealing with long-term translational propagations about small bodies considering orbit–attitude coupled force terms where the attitude is prescribed. Numerical experiments are performed, comparing results to a nominal analytical solution when it is available, or against a highly accurate propagated reference trajectory. The results in this paper show that including the nonconservative forces in the potential energy term for the discrete equations produces a very accurate discretization. This allows one to perform accurate and fast long-term numerical propagations with structure preserving variational algorithms, in scenarios where perturbations to the system can be modeled as nonconservative forces.

Geometric Numerical Integration in Ecological Modelling

A major neglected weakness of many ecological models is the numerical method used to solve the governing systems of differential equations. Indeed, the discrete dynamics described by numerical integrators can provide spurious solution of the corresponding continuous model. The approach represented by the geometric numerical integration, by preserving qualitative properties of the solution, leads to improved numerical behaviour expecially in the long-time integration. Positivity of the phase space, Poisson structure of the flows, conservation of invariants that characterize the continuous ecological models are some of the qualitative characteristics well reproduced by geometric numerical integrators. In this paper we review the benefits induced by the use of geometric numerical integrators for some ecological differential models.

GeCo: Geometric Conservative nonstandard schemes for biochemical systems

We generalize the nonstandard Euler and Heun schemes in order to provide explicit geometric numerical integrators for biochemical systems, here denoted as GeCo schemes, that preserve both positivity of the solutions and linear invariants. We relax the request on the order convergence of the denominator function for the first-order approximation and we let it depend on the step size also throughout the solution approximating values. The first-order variant is exact on a two-dimensional linear test problem. Moreover, we introduce a class of modified mGeCo(α) schemes and, by tuning the parameter α≥1, we improve the numerical performance of GeCo integrators on some examples taken from the literature. A numerical comparison with BBKS and mBBKS schemes, which are implicit integrators, positive and conserve linear invariants, show the gain in efficiency of both GeCo and mGeCo(α) procedures as they generate similar errors with an explicit functional form.

Nearly conservative multivalue methods with extended bounded parasitism

The paper is focused on the analysis of parasitism for multivalue numerical methods intended as geometric numerical integrators for Hamiltonian problems. In particular, the main topic is the design of multivalue numerical methods whose parasitic components remain bounded over certain time intervals, opening the path to the development of nearly conservative multivalue methods able to guarantee a control of parasitism in the long time. The analysis of parasitism as well as the development of the corresponding methods is the core of the treatise. The effectiveness of the approach is also confirmed on selected Hamiltonian problems.

Contact variational integrators

We present geometric numerical integrators for contact flows that stem from a discretization of Herglotz’ variational principle. First we show that the resulting discrete map is a contact transformation and that any contact map can be derived from a variational principle. Then we discuss the backward error analysis of our variational integrators, including the construction of a modified Lagrangian. Throughout the paper we use the damped harmonic oscillator as a benchmark example to compare our integrators to their symplectic analogues.

Modeling and Stability Analysis of Aerially Towed Systems Using Geometric Computational Dynamics

This paper presents a method for three-dimensional simulation and analysis of circularly towed cable–body systems with nontrivial coupling between the attitude dynamics of the towed body and the cable. By application of a Lie group variational integrator, a compact and singularity-free geometric model that maintains its accuracy even for extremely long-time simulations with aggressive maneuvers is developed. Coupling the integrator to a nonvariational model of the towing airplane is also discussed. By employing a variation-based linearization technique, a geometric linear system that is used to investigate the mode shapes and the coupling between the motion of the cable and the attitude dynamics of the towed body is obtained. A distributed parameter model is developed separately that describes the equilibrium configurations of the circularly towed system, the results from which are used to evaluate those of the geometric numerical integrator. For the case of towing a bisymmetric finned body, it is observed that reducing the flight path radius leads to a limit cycle, due to the interactions between the rolling motion of the body and the pendulum mode of the cable.

Totally Conservative Integration Method for the General Three-body Problem and Its Lagrangian Solutions

Abstract In this paper, we design a precise integration method with a variable time step for the general three-body problem that maintains all the conserved quantities. Our method is based on a logarithmic Hamiltonian leapfrog with chain vectors proposed by Mikkola & Tanikawa and features an energy-preserving parameter. Although the proposed method is merely second-order accurate, it can precisely trace some periodic orbits. This is not possible with generic geometric eighth-order numerical integrators and the logarithmic Hamiltonian leapfrog approach. Further, similar to logarithmic Hamiltonian leapfrog, our method is analytically shown to have Lagrangian solutions. Prior to the presented integration method, no integration method was known to preserve all the conserved quantities, in addition to presenting triangular Lagrangian solutions. Because our method is implicit, it requires an iteration method. Therefore, the proposed approach seems to be computationally intensive. However, our method is less computationally burdensome than a generic explicit eighth-order symplectic method.

Computational geometric system identification for the attitude dynamics on SO(3)

A computational approach for system identification for the attitude dynamics of a rigid body is proposed. The estimation and system identification for the rigid body are particularly challenging as they evolve on the compact nonlinear manifold, referred to as the special orthogonal group. Current methods based on local parameterizations or quaternions suffer from inherent singularities or ambiguities associated with them. The proposed method addresses these issues by formulating the system identification problem as an optimization problem on the special orthogonal group. It is solved by a geometric numerical integrator that yields numerical trajectories that are consistent with the geometric structures of Hamiltonian systems on a Lie group. As a result, the proposed method is particularly useful to handle large initial estimation errors that may cause substantial discrepancies between the target attitude trajectories and the initial estimate of them. These are illustrated by numerical examples.

Geometric numerical integrators for Hunter–Saxton-like equations

We present novel geometric numerical integrators for Hunter--Saxton-like equations by means of new multi-symplectic formulations and known Hamiltonian structures of the problems. We consider the Hunter--Saxton equation, the modified Hunter--Saxton equation, and the two-component Hunter--Saxton equation. Multi-symplectic discretisations based on these new formulations of the problems are exemplified by means of the explicit Euler box scheme, and Hamiltonian-preserving discretisations are exemplified by means of the discrete variational derivative method. We explain and justify the correct treatment of boundary conditions in a unified manner. This is necessary for a proper numerical implementation of these equations and was never explicitly clarified in the literature before, to the best of our knowledge. Finally, numerical experiments demonstrate the favourable behaviour of the proposed numerical integrators.

Geometric Numerical Integration for Peakon b-Family Equations

AbstractIn this paper, we study the Camassa-Holm equation and the Degasperis-Procesi equation. The two equations are in the family of integrable peakon equations, and both have very rich geometric properties. Based on these geometric structures, we construct the geometric numerical integrators for simulating their soliton solutions. The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties, however they also have the significant difference, for example there exist the shock wave solutions for the Degasperis-Procesi equation. By using the symplectic Fourier pseudo-spectral integrator, we simulate the peakon solutions of the two equations. To illustrate the smooth solitons and shock wave solutions of the DP equation, we use the splitting technique and combine the composition methods. In the numerical experiments, comparisons of these two kinds of methods are presented in terms of accuracy, computational cost and invariants preservation.

AN EFFICIENT CONSERVATIVE INTEGRATOR WITH A CHAIN REGULARIZATION FOR THE FEW-BODY PROBLEM

We design an efficient orbital integration scheme for the general N-body problem that preserves all the conserved quantities except the angular momentum. This scheme is based on the chain concept and is regarded as an extension of a d'Alembert-type scheme for constrained Hamiltonian systems. It also coincides with the discrete-time general three-body problem for particle number N = 3. Although the proposed scheme is only second-order accurate, it can accurately reproduce some periodic orbits, which cannot be done with generic geometric numerical integrators.

New Langevin and gradient thermostats for rigid body dynamics.

We introduce two new thermostats, one of Langevin type and one of gradient (Brownian) type, for rigid body dynamics. We formulate rotation using the quaternion representation of angular coordinates; both thermostats preserve the unit length of quaternions. The Langevin thermostat also ensures that the conjugate angular momenta stay within the tangent space of the quaternion coordinates, as required by the Hamiltonian dynamics of rigid bodies. We have constructed three geometric numerical integrators for the Langevin thermostat and one for the gradient thermostat. The numerical integrators reflect key properties of the thermostats themselves. Namely, they all preserve the unit length of quaternions, automatically, without the need of a projection onto the unit sphere. The Langevin integrators also ensure that the angular momenta remain within the tangent space of the quaternion coordinates. The Langevin integrators are quasi-symplectic and of weak order two. The numerical method for the gradient thermostat is of weak order one. Its construction exploits ideas of Lie-group type integrators for differential equations on manifolds. We numerically compare the discretization errors of the Langevin integrators, as well as the efficiency of the gradient integrator compared to the Langevin ones when used in the simulation of rigid TIP4P water model with smoothly truncated electrostatic interactions. We observe that the gradient integrator is computationally less efficient than the Langevin integrators. We also compare the relative accuracy of the Langevin integrators in evaluating various static quantities and give recommendations as to the choice of an appropriate integrator.

Geometric numerical integrators based on the Magnus expansion in bifurcation problems for non-linear elastic solids

We illustrate a procedure based on the Magnus expansion for studying mechanical problems which lead to non-autonomous systems of linear ODE’s. The effectiveness of the Magnus method is enlighten by the analysis of a bifurcation problem in the framework of three-dimensional non-linear elasticity. In particular, for an isotropic compressible elastic tube subject to an azimuthal shear primary deformation we study the possibility of axially periodic twist-like bifurcations. The approximate matricant of the resulting differential problem and the first singular value of the bifurcating load corresponding to a non-trivial bifurcation are determined by employing a simplified version of the Magnus method, characterized by a truncation of the Magnus series after the second term.

High-fidelity numerical simulation of complex dynamics of tethered spacecraft

This paper presents an analytical model and a geometric numerical integrator for a tethered spacecraft model that is composed of two rigid bodies connected by an elastic tether. This high-fidelity model includes important dynamic characteristics of tethered spacecraft in orbit, namely the nonlinear coupling among tether deformations, rigid body rotational dynamics, a reeling mechanism, and orbital dynamics. A geometric numerical integrator, referred to as a Lie group variational integrator, is developed to numerically preserve the Hamiltonian structure of the presented model and its Lie group configuration manifold. This approach preserves the geometry of the configurations, and leads to accurate and efficient algorithms that have guaranteed accuracy properties that make them suitable for many dynamic simulations, especially over long simulation times. These analytical and computational models provide a reliable benchmark for testing the validity and applicability of the many simplified models in the existing literature, which have hitherto been used without careful verification that the simplifying assumptions employed are valid in physically realistic parameter regimes. We present numerical simulations which illustrate the important qualitative differences in the tethered spacecraft dynamics when the high-fidelity model is employed, as compared to models with additional simplifying assumptions.

Multisymplectic Fourier pseudo-spectral integrators for Klein-Gordon-Schrödinger equations

A multisymplectic Fourier pseudo-spectral scheme, which exactly preserves the discrete multisym- plectic conservation law, is presented to solve the Klein-Gordon-Schrodinger equations. The scheme is of spectral accuracy in space and of second order in time. The scheme preserves the discrete multisymplectic conservation law and the charge conservation law. Moreover, the residuals of some other conservation laws are derived for the geometric numerical integrator. Extensive numerical simulations illustrate the numerical behavior of the multisymplectic scheme, and demonstrate the correctness of the theoretical analysis.