Journal of Computational Dynamics
December 2019 , Volume 6 , Issue 2
Special issue in honor of Reinout Quispel
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The objective of this work is the introduction and investigation of favourable time integration methods for the Gross-Pitaevskii equation with rotation term. Employing a reformulation in rotating Lagrangian coordinates, the equation takes the form of a nonlinear Schrödinger equation involving a space-time-dependent potential. A natural approach that combines commutator-free quasi-Magnus exponential integrators with operator splitting methods and Fourier spectral space discretisations is proposed. Furthermore, the special structure of the Hamilton operator permits the design of specifically tailored schemes. Numerical experiments confirm the good performance of the resulting exponential integrators.
We consider recent work of [
Aromatic B-series are a generalization of B-series. Some of the algebraic structures on B-series can be defined analogically for aromatic B-series. This paper derives combinatorial formulas for the composition and substitution laws for aromatic B-series.
A parameter-dependent class of Hamiltonian (generalized) Lotka–Volterra systems is considered. We prove that this class contains Liouville integrable as well as superintegrable cases according to particular choices of the parameters. We determine sufficient conditions which result in integrable behavior, while we numerically explore the complementary cases, where these analytically derived conditions are not satisfied.
We explore a new asymptotic-numerical solver for the time-dependent wave equation with an interaction term that is oscillating in time with a very high frequency. The method involves representing the solution as an asymptotic series in inverse powers of the oscillation frequency. Using the new scheme, high accuracy is achieved at a low computational cost. Salient features of the new approach are highlighted by a numerical example.
Principal Geodesic Analysis is a statistical technique that constructs low-dimensional approximations to data on Riemannian manifolds. It provides a generalization of principal components analysis to non-Euclidean spaces. The approximating submanifolds are geodesic at a reference point such as the intrinsic mean of the data. However, they are local methods as the approximation depends on the reference point and does not take into account the curvature of the manifold. Therefore, in this paper we develop a specialization of principal geodesic analysis, Principal Symmetric Space Analysis, based on nested sequences of totally geodesic submanifolds of symmetric spaces. The examples of spheres, Grassmannians, tori, and products of two-dimensional spheres are worked out in detail. The approximating submanifolds are geometrically the simplest possible, with zero exterior curvature at all points. They can deal with significant curvature and diverse topology. We show that in many cases the distance between a point and the submanifold can be computed analytically and there is a related metric that reduces the computation of principal symmetric space approximations to linear algebra.
In this paper we construct a family of integrable reductions of the dressing chain, described in its Lotka-Volterra form. For each
Finite difference schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic nonlinearity.
In principle, a simplified version of the direct approach also works for equations with polynomial nonlinearity of higher degree. For the modified Korteweg-de Vries equation, whose nonlinear term is cubic, this approach yields several new families of second-order accurate schemes that preserve mass and either energy or momentum. Two of these families contain Average Vector Field schemes of the type developed by Quispel and co-workers. Numerical tests show that each family includes schemes that are highly accurate compared to other mass-preserving methods that can be found in the literature.
A QRT map is the composition of two involutions on a biquadratic curve: one switching the
Classical mechanical systems are defined by their kinetic and potential energies. They generate a Lie algebra under the canonical Poisson bracket. This Lie algebra, which is usually infinite dimensional, is useful in analyzing the system, as well as in geometric numerical integration. But because the kinetic energy is quadratic in the momenta, the Lie algebra obeys identities beyond those implied by skew symmetry and the Jacobi identity. Some Poisson brackets, or combinations of brackets, are zero for all choices of kinetic and potential energy, regardless of the dimension of the system. Therefore, we study the universal object in this setting, the 'Lie algebra of classical mechanics' modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket. We show that it is the direct sum of an abelian algebra
We propose a Fourier pseudo-spectral scheme for the space-fractional nonlinear Schrödinger equation. The proposed scheme has the following features: it is linearly implicit, it preserves two invariants of the equation, its unique solvability is guaranteed without any restrictions on space and time step sizes. The scheme requires solving a complex symmetric linear system per time step. To solve the system efficiently, we also present a certain variable transformation and preconditioner.
In this paper, we exploit the gradient flow structure of continuous-time formulations of Bayesian inference in terms of their numerical time-stepping. We focus on two particular examples, namely, the continuous-time ensemble Kalman–Bucy filter and a particle discretisation of the Fokker–Planck equation associated to Brownian dynamics. Both formulations can lead to stiff differential equations which require special numerical methods for their efficient numerical implementation. We compare discrete gradient methods to alternative semi-implicit and other iterative implementations of the underlying Bayesian inference problems.
Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field with a linear Poisson tensor and with a quadratic Hamilton function, this map is known to be integrable and to preserve a pencil of conics. In the paper "Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation" by P. van der Kamp et al. [
We consider the dynamics and numerical simulation of systems of linked rigid bodies (chains). We describe the system using the moving frame method approach of [
In plasma simulations, numerical methods with high computational efficiency and long-term stability are needed. In this paper, symplectic methods with adaptive time steps are constructed for simulating the dynamics of charged particles under the electromagnetic field. With specifically designed step size functions, the motion of charged particles confined in a Penning trap under three different magnetic fields is studied, and also the dynamics of runaway electrons in tokamaks is investigated. The numerical experiments are performed to show the efficiency of the new derived adaptive symplectic methods.
We conjecture recurrence relations satisfied by the degrees of some linearizable lattice equations. This helps to prove linear degree growth of these equations. We then use these recurrences to search for lattice equations that have linear growth and hence are linearizable.
A system of ordinary differential equations of a predator–prey type, depending on nine parameters, is studied. We have included in this model a nonmonotonic response function and time periodic perturbation. Using numerical continuation software, we have detected three codimension two bifurcations for the unperturbed system, namely cusp, Bogdanov-Takens and Bautin bifurcations. Furthermore, we concentrate on two regions in the parameter space, the region where the Bogdanov-Takens and the region where Bautin bifurcations occur. As we turn on the time perturbation, we find strange attractors in the neighborhood of invariant tori of the unperturbed system.
The theory of moving frames has been used successfully to solve one dimensional (1D) variational problems invariant under a Lie group symmetry. In the one dimensional case, Noether's laws give first integrals of the Euler–Lagrange equations. In higher dimensional problems, the conservation laws do not enable the exact integration of the Euler–Lagrange system. In this paper we use the theory of moving frames to help solve, numerically, some higher dimensional variational problems, which are invariant under a Lie group action. In order to find a solution to the variational problem, we need first to solve the Euler Lagrange equations for the relevant differential invariants, and then solve a system of linear, first order, compatible, coupled partial differential equations for a moving frame, evolving on the Lie group. We demonstrate that Lie group integrators may be used in this context. We show first that the Magnus expansions on which one dimensional Lie group integrators are based, may be taken sequentially in a well defined way, at least to order 5; that is, the exact result is independent of the order of integration. We then show that efficient implementations of these integrators give a numerical solution of the equations for the frame, which is independent of the order of integration, to high order, in a range of examples. Our running example is a variational problem invariant under a linear action of
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