On high order symmetric and symplectic trigonometrically fitted Runge-Kutta methods with an even number of stages

Abstract

The existence and construction of symplectic 2s-stage variable coefficients Runge-Kutta (RK) methods that integrate exactly IVPs whose solution is a trigonometrical polynomial of order s with a given frequency ω is considered. The resulting methods, that can be considered as trigonometrical collocation methods, are fully implicit, symmetric and symplectic RK methods with variable nodes and coefficients that are even functions of ν = ωh (h is the step size), and for ω → 0 they tend to the conventional RK Gauss methods. The present analysis extends previous results on two-stage symplectic exponentially fitted integrators of Van de Vyver (Comput. Phys. Commun. 174: 255-262, 2006) and Calvo et al. (J. Comput. Appl. Math. 218: 421-434, 2008) to symmetric and symplectic trigonometrically fitted methods of high order. The algebraic order of the trigonometrically fitted symmetric and symplectic 2s-stage methods is shown to be 4s like in conventional RK Gauss methods. Finally, some numerical experiments with oscillatory Hamiltonian systems are presented.

Symmetric and symplectic exponentially fitted Runge–Kutta methods of high order

“…(iii) The results stated in Lemmas 2.4 and 2.5 indicate that for symmetric FFRK methods there is an important reduction of the fitting conditions (27), and this reduction is collected in the following Corollary 2.6. For an s-stage symmetric FFRK method fitted to a linear space F ⊂ H 1 ∪ H 2 , the fitting conditions (27) …”

“…When the frequency used in the exponential fitting process is λ = 0 (z = 0), the new integrators reduce to the standard four-stage eighth-order Gauss integrator. It is shown that such fitted methods are a reliable alternative to the standard four-stage eighth-order Gauss integrator and the trigonometrically fitted four-stage eighth-order STFRK4 code derived in [27] to describe the evolution of some oscillatory problems. Furthermore, the computational cost of the new modified EFRK methods is similar to their counterparts standard RK methods.…”

“…For an s-stage symmetric FFRK method fitted to a linear space F ⊂ H 1 ∪ H 2 , the fitting conditions (27) …”

“…The new methods have been compared with the standard four-stage eighth-order Gauss method given in [2] denoted as Gauss4, and with the four-stage, eighth-order variable-nodes integrator STFRK4 derived in [27]. It is the symplectic, symmetric and trigonometrically fitted to 1, cos(ωt), sin(ωt), .…”

Functionally Fitted Energy-Preserving Methods for Solving Oscillatory Nonlinear Hamiltonian Systems

“…Just as the research of symplectic and EP methods, EF/TF symplectic methods have been studied extensively by many authors (see, e.g. [7,8,9,15,36,37,40]). By contrast, as far as we know, only a few papers paid attention to the EF/TF EP methods (see, e.g.…”