A note on symplectic and symmetric ARKN methods☆
Introduction
Consider the separable Hamiltonian systems of the form: where and are generalized positions and generalized momenta, respectively. is the gradient of a real-valued function whose second derivatives are continuous and the Hamiltonian of the system is . We note that this Hamiltonian system is identical to the following oscillatory system with .
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 the corresponding ARKN methods for (2) are formulated by where are even functions of , and 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 and of an ARKN method are nonzero.
It follows from conditions (4) that then
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 , a symplectic ARKN integrator reduces to a classical symplectic RKN method with the same algebraic order.
References (7)
Comput. Phys. Comm.
(2002)- et al.
Comput. Phys. Comm.
(2010) - et al.
Comput. Phys. Comm.
(2009)
Cited by (7)
The existence of explicit symplectic ARKN methods with several stages and algebraic order greater than two
2019, Journal of Computational and Applied MathematicsSymplectic and symmetric trigonometrically-fitted ARKN methods
2019, Applied Numerical MathematicsCitation Excerpt :Moreover, Franco [7] and Wu et al. [27] extended ARKN integrators to multidimensional systems. For more work we see [16,17,26]. Recently, in order to let ARKN methods behave better in integrating oscillatory problems (1), Yang et al. [28] modified the ARKN methods by introducing frequency depending coefficients into the terms in the internal stages and proposed trigonometrically-fitted ARKN (TFARKN) methods.
Oscillation-preserving algorithms for efficiently solving highly oscillatory second-order ODEs
2021, Numerical AlgorithmsGeometric Integrators for Differential Equations with Highly Oscillatory Solutions
2021, Geometric Integrators for Differential Equations with Highly Oscillatory SolutionsRecent developments in structure-preserving algorithms for oscillatory differential equations
2018, Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations
- ☆
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.