A note on continuous-stage Runge–Kutta methods

Abstract

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...

Energy-preserving continuous-stage partitioned Runge-Kutta methods

“…Example 3.4. By taking η = s and α (i,j) = 0, α (i,j) = 0 for i, j ≥ η in Theorem 3.5, we regain the class of energy-preserving methods which are symmetric, conjugate-symplectic up to order 2s + 2 and have a super-convergence order 2s [22,36,41]. Such methods coincide with the limit form of Hamiltonian boundary value methods (denoted by HBVM(∞, s)) [5], the s-degree continuous time finite element methods [34] and the optimal order energy-preserving variant of collocation methods [21].…”

“…Actually, it can be addressed by using the orthogonal polynomial expansion technique in conjunction with the order conditions. One possible way is that we can use the same approach as presented in [40,41] for constructing methods of arbitrary order, i.e., substituting (2.10) into the order conditions 3 one by one and determining the corresponding parameters α (i,j) (see Theorem 3.1 below as a simple example). However, such a approach may lead to increasing-complicated computations when the order goes much higher, hence it may be not suitable for devising high-order methods and for this reason we do not plan to pursue it here.…”

“…For ease of employing Theorem 3.2, hereafter we always assume C τ = C τ = τ which is a natural assumption used in the previous studies [36,41,42]. then we have B τ = B τ = 1 and moreover, the method is of order at least 2.…”

“…In this paper, we are interested in the continuous-stage approaches for numerical discretization of ordinary differential equations, the seminal idea of which were introduced by Butcher in 1972 [7] (see also [8,9]) and subsequently developed by Hairer in 2010 [21]. We mention some typical applications of such approaches in the study of geometric integration as follows: there are several existing energypreserving integrators that can be connected to Runge-Kutta (RK) methods with continuous stage [5,10,13,21,25,26,27,28,34,47]; the conjugate-symplecticity of energy-preserving methods can be discussed in the context of continuous-stage Runge-Kutta (csRK) methods [21,22,36]; both symplectic and symmetric integrators can be devised in use of the notions of Runge-Kutta (RK), partitioned Runge-Kutta (PRK) and Runge-Kutta-Nyström (RKN) methods with continuous stage [35,36,37,38,40,41,42,43,44,45,48]; it is known that some symplectic integrators derived from Galerkin variational problems can be interpreted and analyzed in the framework of continuous-stage partitioned Runge-Kutta (csPRK) methods [34,39,46]. Particularly, it is worth mentioning that energy-preserving integrators can be easily constructed by using csRK approaches [26,27,35,36], which result in energy-preserving RK methods by using quadrature formulas.…”

A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications

“…Around 2000, Bridges [17] and Reich [18] first put forward the multisymplectic algorithms. In recent years, the meshless symplectic algorithms [19,20], the symplectic continuous-stage Runge-Kutta methods [21,22], the Fourier spectral/ pseudospectral methods [23,24], and other symplectic algorithms have been developed in succession. A large number of numerical simulations indicate that the symplectic algorithms have superiority in conservation and long-term tracking ability.…”