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The potential for adaptive explicit Runge–Kutta (ERK) codes to produce global errors that decrease linearly as a function of the error tolerance is studied. It is shown that this desirable property may not hold, in general, if the leading term of the locally computed error estimate passes through zero. However, it is also shown that certain methods are… (More)

We present and analyze energy-conserving methods for the numerical integration of IVPs of Poisson type systems that are able to preserve some Casimirs. Their derivation and analysis is done within the framework following the ideas of Hamiltonian BVMs (HBVMs) (see [1] and references therein). The proposed methods turn out to be equivalent to those recently… (More)

The so-called multi-revolution methods were introduced in celestial mechanics as an efficient tool for the long-term numerical integration of nearly periodic orbits of artificial satellites around the Earth. A multi-revolution method is an algorithm that approximates the map ϕ N T of N near-periods T in terms of the one near-period map ϕT evaluated at few s… (More)

a r t i c l e i n f o a b s t r a c t The construction of high order symmetric, symplectic and exponentially fitted Runge–Kutta (RK) methods for the numerical integration of Hamiltonian systems with oscillatory solutions is analyzed. Based on the symplecticness, symmetry, and exponential fitting properties, three new four-stage RK integrators, either with… (More)