Symplectic and Symmetric Multidimensional ERKN Methods
Symplectic and symmetric Runge–Kutta–Nyström-type methods have been shown to have excellent behavior in the long-term integration of Hamiltonian systems. Chapter 4 focuses on the investigation of symplectic and symmetric multi-frequency and multidimension
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Symplectic and Symmetric Multidimensional ERKN Methods
This chapter is devoted to the investigation of symplectic and symmetric multidimensional extended Runge–Kutta–Nyström (SSMERKN) integrators. The symplecticity and symmetry conditions for multidimensional ERKN methods are obtained. When the principal frequency matrix vanishes, they reduce to those for the traditional RKN methods with constant coefficients. Some practical SSMERKN integrators are derived. The stability and phase properties of SSMERKN integrators are analyzed. Finally, we present a technique for transforming a non-autonomous Hamiltonian system into an equivalent autonomous Hamiltonian system in an extended phase space. Symplectic multidimensional ERKN methods applied to the equivalent system are shown to preserve the extended energy very well.
4.1 Symplecticity and Symmetry Conditions for Multidimensional ERKN Integrators Conservative mechanical systems are time-reversible in the sense that reversing the initial direction of the velocity vector at a certain position results in a motion in the opposite direction along the same trajectory. This structure of the system is wellpreserved by symmetric integrators (see Chap. V of Hairer et al. [20]). On the other hand, many problems in applications take the form of a Hamiltonian system p˙ = −∇q H (p, q),
q˙ = ∇p H (p, q),
where H : Rd × Rd → R, and the dimension d is the number of degrees of freedom. It is well-known that the solution of the system preserves the Hamiltonian H (p, q): dtd H (p(t), q(t)) = 0, and the corresponding flow is symplectic in the sense that the differential 2-form ω2 = dp ∧ d q is invariant: dtd dp ∧ dq = 0. The important property of symplecticity is well-preserved by the so-called symplectic integrators. See Sanz-Serna et al.’s monograph [34], Chap. VI of Hairer et al.’s X. Wu et al., Structure-Preserving Algorithms for Oscillatory Differential Equations, DOI 10.1007/978-3-642-35338-3_4, © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2013
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Symplectic and Symmetric Multidimensional ERKN Methods
book [20], and Leimkuhler and Reich [26] for a systematic survey of symplectic methods for Hamiltonian systems. For other references we refer the reader to [2–4, 7, 35, 38, 42, 52]. Consider q¨ + Mq = f (q), t ∈ [t0 , tend ], (4.1) q(t0 ) = q0 , q(t ˙ 0 ) = p0 , where q : R → Rd and M ∈ Rd×d is a symmetric and positive semi-definite matrix implicitly containing the frequencies of the problem. When f (q) = −∇U (q) for some continuously differentiable function U (q), (4.1) is equivalent to a separable Hamiltonian system with the Hamiltonian H (p, q) = 12 p T p + 12 q T Mq + U (q). Since symplectic and symmetric methods have excellent structure-preserving behavior in long-term integration, it is natural to require the numerical integrators for the problem (4.1) proposed in Chap. 3 to be symmetric and symplectic.
4.1.1 Symmetry Conditions A symmetric method aims at producing a reversible numerical flow when applied to a reversible differential equation. As me
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