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

Abstract

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.

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

“…In recent years, numerical methods with infinitely many stages including continuous-stage Runge-Kutta (csRK) methods, continuous-stage partitioned Runge-Kutta (csPRK) methods and continuous-stage Runge-Kutta-Nyström (csRKN) methods are presented and discussed by several authors, see [1,16,18,20,21,26,27,28,29,31,32,33,34,35]. They can be viewed as the natural generalizations of numerical methods with finite stages (e.g., classical RK methods).…”

“…They can be viewed as the natural generalizations of numerical methods with finite stages (e.g., classical RK methods). It is shown in [28,29,31,32,33,34,35] that by using continuous-stage methods many classical RK, PRK and RKN methods of arbitrary order can be derived, without resort to solving the tedious nonlinear algebraic equations (associated with order conditions) in terms of many unknown coefficients.…”

“…The construction of continuous-stage methods seems much easier than that of those traditional methods with finite stages, as the associated Butcher coefficients are "continuous" or "smooth" functions and hence they can be treated by using some analytical tools [28,29,31,32,33,34,35]. Moreover, as presented in [4,16,18,20,21,28,29,31,32,33,34,35], numerical methods serving some special purpose including symplecticity-preserving methods for Hamiltonian systems, symmetric methods for reversible systems, energy-preserving methods for conservative systems can also be established within this new framework. Besides, a well known negative result we have to mention here is that no RK methods is energy-preserving for general non-polynomial Hamiltonian systems [8], in contrast to this, energy-preserving csRK methods can be easily constructed [1,2,16,18,23,26,27,28,20].…”

“…This paper will be organized as follows. In Section 2, we introduce the exact definition of csRKN methods for solving second-order ODEs and the corresponding order theory previously developed in [32] will be briefly revisited. In Section 3, by using Legendre expansion technique, we present some useful results for devising symmetric integrators which is then followed by giving some illustrative examples for deriving new symmetric integrators in Section 4.…”

A note on the continuous-stage Runge–Kutta(–Nyström) formulation of Hamiltonian Boundary Value Methods (HBVMs)

“…Also in this case, one could derive the result through the simplifying assumptions for continuous-stage RKN methods[37].…”