Symplecticity-preserving continuous-stage Runge–Kutta–Nyström methods

Sun

Zhang

2019

Self Cite

In this article, we develop high-order symplectic integrators for solving second order differential equations which can be transformed into separable Hamiltonian systems. The construction of such high-order integrators is based on the notion of continuous-stage Runge-Kutta-Nyström methods in conjunction with the Legendre polynomial expansion techniques and simplifying assumptions of order conditions. As examples, three new one-parameter families of symplectic methods which are of order 4, 6 and 8 respectively are derived in use of Gaussian-type quadrature. Some numerical tests are well performed to verify our theoretical results.

“…where we have interchanged the indexes i and j. Substituting the above two expressions into (36), it yields…”

Section: Symplectic Conditions For Csrkn Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Sun

Zhang

2019

Self Cite

“…with one parameter α being introduced, then we get a family of symmetric csRKN methods with order 2. By Theorem 3.3 presented in [31] (see also Theorem 4.4 in [32]), such methods are also symplectic and thus suitable for solving general second-order Hamiltonian systems. By using suitable quadrature formulas with order p ≥ 2 we can get symmetric RKN methods of order 2 2.…”

Section: Symmetric Rkn Methodsmentioning

confidence: 98%

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

Zhang

2019

Self Cite

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-Nyström 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.

“…Instead of using Lagrangian polynomials [19], Tang et al [37,38,39,44] resort to Legendre orthogonal polynomials for constructing csRK methods, and some special-purpose methods including symplectic csRK methods [38,39,44], conjugate-symplectic (up to a finite order) csRK methods [39], symmetric csRK methods [38,39,44], energy-preserving csRK methods [38,39,44] are developed in use of Legendre polynomials. Besides, Some extensions of csRK methods are also obtained by Tang et al [40,41,43]. Another interesting related work are given by Li & Wu [25], showing that their energy-preserving methods can be explained as a class of csRK methods.More recently, new ideas have been introduced for construction of csRK methods in [44,45,46,47], where the Butcher coefficients are partially assumed to be polynomial functions by…”

mentioning

confidence: 97%

A note on continuous-stage Runge–Kutta methods

2018

Self Cite

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 symmetric and symplectic integrators by using Legendre polynomials, Laguerre polynomials and Hermite polynomials are constructed. significant importance mainly owing to that they are more convenient to use and sometimes it may lead to surprising applications [6].The well-known Runge-Kutta (RK) method, as one of the most popular classes of methods for solving initial value problems, has been a central topic in numerical ordinary differential equations (ODEs) since the pioneering work of Runge (1895) [29]. They have been highly developed for over a hundred and twenty years (see [7,4,8,16,17] and references therein). One of the most important points we have to speak out is that RK methods possess a close relationship with polynomials. A basic evidence for supporting such view is that, the RK order conditions corresponding to bushy trees (by B-series theory [8,18]) are equivalent to those polynomial-based quadrature rules [16]. Another good case in evidence is the Wtransformation [17] which is defined on the basis of Legendre orthogonal polynomials. By using W -transformation many special-purpose methods can be established, e.g., symplectic RK methods, symmetric RK methods, algebraically stable RK methods, stiffly accurate & L-stable RK methods etc [10,11,16,17,24,33,34].A novel RK approach reflecting stronger relationship between RK-type methods and polynomials are developed in recent years. It is clear and easy to understand it by introducing the concept of "continuous-stage Runge-Kutta (csRK) methods". Actually, the theory of csRK methods was initially launched by Butcher in 1972 [5] (see also [8]) and subsequently developed by Hairer [19,20], Tang & Sun [37,38,39,42], and Miyatake & Butcher [26,27]. The coefficients of csRK methods are assumed to be "continuous" functions and thus it is a simple and natural choice to let them be polynomials. The first example of such methods was given by Hairer using Lagrangian interpolatory polynomials for deriving energy-preserving collocation (EPC) methods [19], whereas a low-order version of EPC methods was proposed earlier by Quispel & McLaren [28] with the name "averaged vector field (AVF) methods" but without being interpreted as csRK methods. A closely-related type of methods which can also be transformed into csRK methods attributes to Brugnano et al [3], called infinity Hamiltonian Boundary Value Methods (denoted by ∞-HBVMs), but also had not been explained within the framework of csRK methods at that time. It was firstly pointed out in [37] (and subsequentl...