Energy-preserving integration of non-canonical Hamiltonian systems by continuous-stage methods

As is well known, energy is generally deemed as one of the most important physical invariants in many conservative problems and hence it is of remarkable interest to consider numerical methods which are able to preserve it. In this paper, we are concerned with the energy-preserving integration of non-canonical Hamiltonian systems by continuous-stage methods. Algebraic conditions in terms of the Butcher coefficients for ensuring the energy preservation, symmetry and quadratic-Casimir preservation respectively are presented. With the presented condition and in use of orthogonal expansion techniques, the construction of energy-preserving integrators is examined. A new class of energy-preserving integrators which is symmetric and of order $2m$ is constructed. Some numerical results are reported to verify our theoretical analysis and show the effectiveness of our new methods

Two types of variational integrators and their equivalence

In this paper, we introduce two types of variational integrators, one originating from the discrete Hamilton's principle while the other from Galerkin variational approach. It turns out that these variational integrators are equivalent to each other when they are used for integrating the classical mechanical system with Lagrangian function $L(q,\dot{q})=\frac{1}{2}\dot{q}^TM\dot{q}-U(q)$ ($M$ is an invertible symmetric constant matrix). They are symplectic, symmetric, possess super-convergence order $2s$ (which depends on the degree of the approximation polynomials), and can be related to continuous-stage partitioned Runge-Kutta methods

An extended framework of continuous-stage Runge-Kutta methods

We propose an extended framework for continuous-stage Runge-Kutta methods which enables us to treat more complicated cases especially for the case weighting on infinite intervals. By doing this, various types of weighted orthogonal polynomials (e.g., Jacobi polynomials, Laguerre polynomials, Hermite polynomials etc.) can be used in the construction of Runge-Kutta-type methods. Particularly, families of Runge-Kutta-type methods with geometric properties can be constructed in this new framework. As examples, some new symplectic integrators by using Legendre polynomials, Laguerre polynomials and Hermite polynomials are constructed.Comment: arXiv admin note: text overlap with arXiv:1806.0338

Symplecticity-preserving continuous-stage Runge-Kutta-Nystr\"{o}m methods

We develop continuous-stage Runge-Kutta-Nystr\"{o}m (csRKN) methods for solving second order ordinary differential equations (ODEs) in this paper. The second order ODEs are commonly encountered in various fields and some of them can be reduced to the first order ODEs with the form of separable Hamiltonian systems. The symplecticity-preserving numerical algorithm is of interest for solving such special systems. We present a sufficient condition for a csRKN method to be symplecticity-preserving, and by using Legendre polynomial expansion we show a simple way to construct such symplectic RKN type method.Comment: 23 page

Symmetric integrators based on continuous-stage Runge-Kutta-Nystrom methods for reversible systems

In this paper, we study symmetric integrators for solving second-order ordinary differential equations on the basis of the notion of continuous-stage Runge-Kutta-Nystrom methods. The construction of such methods heavily relies on the Legendre expansion technique in conjunction with the symmetric conditions and simplifying assumptions for order conditions. New families of symmetric integrators as illustrative examples are presented. For comparing the numerical behaviors of the presented methods, some numerical experiments are also reported

High order symplectic integrators based on continuous-stage Runge-Kutta Nystrom methods

On the basis of the previous work by Tang \& Zhang (Appl. Math. Comput. 323, 2018, p. 204--219), in this paper we present a more effective way to construct high-order symplectic integrators for solving second order Hamiltonian equations. Instead of analyzing order conditions step by step as shown in the previous work, the new technique of this paper is using Legendre expansions to deal with the simplifying assumptions for order conditions. With the new technique, high-order symplectic integrators can be conveniently devised by truncating an orthogonal series

Construction of symplectic (partitioned) Runge-Kutta methods with continuous stage

Hamiltonian systems are one of the most important class of dynamical systems with a geometric structure called symplecticity and the numerical algorithms which can preserve such geometric structure are of interest. In this article we study the construction of symplectic (partitioned) Runge-Kutta methods with continuous stage, which provides a new and simple way to construct symplectic (partitioned) Runge-Kutta methods in classical sense. This line of construction of symplectic methods relies heavily on the expansion of orthogonal polynomials and the simplifying assumptions for (partitioned) Runge-Kutta type methods.Comment: 13 page

Energy-preserving continuous-stage Runge-Kutta-Nystr\"om methods

Many practical problems can be described by second-order system $\ddot{q}=-M\nabla U(q)$, in which people give special emphasis to some invariants with explicit physical meaning, such as energy, momentum, angular momentum, etc. However, conventional numerical integrators for such systems will fail to preserve any of these quantities which may lead to qualitatively incorrect numerical solutions. This paper is concerned with the development of energy-preserving continuous-stage Runge-Kutta-Nystr\"om (csRKN) methods for solving second-order systems. Sufficient conditions for csRKN methods to be energy-preserving are presented and it is proved that all the energy-preserving csRKN methods satisfying these sufficient conditions can be essentially induced by energy-preserving continuous-stage partitioned Runge-Kutta methods. Some illustrative examples are given and relevant numerical results are reported

Chebyshev symplectic methods based on continuous-stage Runge-Kutta methods

We develop Chebyshev symplectic methods based on Chebyshev orthogonal polynomials of the first and second kind separately in this paper. Such type of symplectic methods can be conveniently constructed with the newly-built theory of weighted continuous-stage Runge-Kutta methods. A few numerical experiments are well performed to verify the efficiency of our new methods.Comment: arXiv admin note: text overlap with arXiv:1805.0995

Symplectic integration with Jacobi polynomials

In this paper, we study symplectic integration of canonical Hamiltonian systems with Jacobi polynomials. The relevant theoretical results of continuous-stage Runge-Kutta methods are revisited firstly and then symplectic methods with Jacobi polynomials will be established. A few numerical experiments are well performed to verify the efficiency of our new methods.Comment: arXiv admin note: text overlap with arXiv:1805.1123