Construction of Runge–Kutta type methods for solving ordinary differential equations

Tang, Wensheng; Sun, Yajuan

doi:10.1016/j.amc.2014.02.042

Cited by 42 publications

(83 citation statements)

References 19 publications

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“…According to Proposition 2.1 shown in [28], we assume B τ = 1 in the current paper and to proceed conveniently, hereafter we also assume C τ = τ . In such a case, B(ξ) is reduced to…”

Section: Continuous-stage Rk Methodsmentioning

confidence: 99%

“…Runge-Kutta (RK) method with continuous stage has been discussed and investigated by several authors recently [15,25,26,28,18], the idea of which was firstly presented by Butcher in 1970s [4,6]. There are few literatures involving this kind of methods since Butcher's seminal work, until recently, [15] exploits it to interpret the energy-preserving collocation methods.…”

Section: Introductionmentioning

confidence: 99%

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

Tang

Lang

Luo

2016

Applied Mathematics and Computation

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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.

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“…According to Proposition 2.1 shown in [28], we assume B τ = 1 in the current paper and to proceed conveniently, hereafter we also assume C τ = τ . In such a case, B(ξ) is reduced to…”

Section: Continuous-stage Rk Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Tang

Lang

Luo

2016

Applied Mathematics and Computation

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“…Tang and Sun have also provided essentially the same sufficient condition in a different manner [22].…”

Section: Sufficient Condition Theorem 31 ([17]) a Csrk Methods Is Ementioning

confidence: 98%

A Characterization of Energy-Preserving Methods and the Construction of Parallel Integrators for Hamiltonian Systems

Miyatake

Butcher

2016

SIAM J. Numer. Anal.

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“…If we shift the Legendre polynomials from [−1, 1] to the standard interval [0, 1], then we can use Theorem 4.3 and 4.10 to derive many classical high-order symmetric and symplectic RK methods (e.g., Gauss-Legendre RK methods). The construction of symplectic RK methods with shifted Legendre polynomials has been investigated in the literature [38,39,41,44]. Let ξ = 3, η = 1, ρ = 2, after some elementary calculations, it gives…”

Section: Some Examplesmentioning

confidence: 99%

A note on continuous-stage Runge–Kutta methods

Tang

2018

Applied Mathematics and Computation

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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...

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Construction of Runge–Kutta type methods for solving ordinary differential equations

Cited by 42 publications

References 19 publications

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

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

A Characterization of Energy-Preserving Methods and the Construction of Parallel Integrators for Hamiltonian Systems

A note on continuous-stage Runge–Kutta methods

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