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Practical symplectic partitioned Runge-Kutta and Runge-Kutta-Nyström methods. (English) Zbl 1001.65078

Summary: We present new symmetric fourth and sixth-order symplectic partitioned Runge-Kutta and Runge-Kutta-Nyströom methods. We studied compositions using several extra stages, optimising the efficiency. An effective error, \(E_f\), is defined and an extensive search is carried out using the extra parameters. The new methods have smaller values of \(E_f\) than other methods found in the literature. When applied to several examples they perform up to two orders of magnitude better than previously known method, which is in very good agreement with the values of \(E_f\).

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A26 Geometric methods in ordinary differential equations
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