A note on symplectic and symmetric ARKN methods

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Abstract

For dealing with the oscillatory separable Hamiltonian system of the form q(t)+ω2q(t)=f(q(t)) with ω(>0), the main frequency of the system, the symplecticity conditions of ARKN methods and the existence of symmetric ARKN methods are investigated by Shi et al. [W. Shi, X. Wu, Comput. Phys. Comm. 183 (2012) 1250]. However, the symplecticity conditions in that paper are not sufficient, one of the conditions should be made stronger to guarantee the symplecticity of the methods. For this reason, we will reanalyze and present more precisely the symplecticity conditions for the ARKN methods in this note.

Introduction

Consider the separable Hamiltonian systems of the form: {q=p,p=V(q),q(t0)=q0,p(t0)=p0, where q:RRd and p:RRd are generalized positions and generalized momenta, respectively. V(q) is the gradient of a real-valued function V whose second derivatives are continuous and the Hamiltonian of the system is H(p,q)=12pTp+V(q). We note that this Hamiltonian system is identical to the following oscillatory system {q(t)+ω2q(t)=f(q(t)),t[t0,T],q(t0)=q0,q(t0)=q0 with q=p,f(q)ω2q=V(q).

For this kind of oscillatory second-order initial value problems, adapted Runge–Kutta–Nyström methods (ARKN methods) were investigated by several authors (see  [1], [2], [3]).

As one of the important qualitative properties of Hamiltonian systems, symplecticity has been receiving more and more attention. Many symplectic or canonical integrators have been proposed. The symplecticity conditions of ARKN methods are investigated in  [4], [5]. However, the symplecticity conditions in  [4], [5] are not sufficient and one of the conditions should be made stronger to make sure the symplecticness of the ARKN methods. This motivates us to give more precise symplecticity conditions for the methods in this note.

Section snippets

ARKN methods and the corresponding symplecticity conditions

Taking account of the special structure brought by the linear term ω2q the corresponding ARKN methods for (2) are formulated by {Qi=qn+cihqn+h2j=1sāij(f(Qj)ω2Qj),i=1,2,,s,qn+1=ϕ0(v)qn+hϕ1(v)qn+h2i=1sb̄i(v)f(Qi),qn+1=ωvϕ1(v)qn+ϕ0(v)qn+hi=1sbi(v)f(Qi), where v=ωh,b̄i(v),bi(v),i=1,2,,s are even functions of v, and ϕ0(v)=cosv,ϕ1(v)=sinvv are Bettis’ functions.

The symplecticity conditions for the ARKN methods are represented by the following theorem with a complete analysis.

Theorem 2.1

(i)  If the

The Butcher tableau of symplectic ARKN methods

In this section, we will reconsider the expression of Butcher tableau for symplectic ARKN methods. With the revised symplecticity conditions given in Theorem 2.1, some slight changes in the results of Section 4 in the original publication  [4] should be made. To this end, we first assume that all the weights bi and b̄i of an ARKN method are nonzero.

It follows from conditions (4) that b̄i=ϕ1(v)ciϕ0(v)ϕ0(v)+civ2ϕ1(v)bi, then biϕ0(v)+v2b̄iϕ1(v)=biϕ0(v)+v2ϕ1(v)ciϕ0(v)ϕ0(v)+civ2ϕ1(v)ϕ1(v)bi=biϕ0(v)

Conclusions

In this note, we represent precise symplecticity conditions for ARKN methods and give the revised Butcher tableau of symplectic ARKN methods. It also can be observed that the symplecticity conditions for ARKN methods contain the symplecticity conditions for classical RKN methods, and when v0, a symplectic ARKN integrator reduces to a classical symplectic RKN method with the same algebraic order.

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The research is supported in part by the Natural Science Foundation of China under Grant 11271186, by the Specialized Research Foundation for the Doctoral Program of Higher Education under Grant 20100091110033, by the 985 Project at Nanjing University under Grant 9112020301, by a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and by the University Postgraduate Research and Innovation Project of Jiangsu Province 2012 under Grant CXLX12_0033.

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