Derivation and implementation of Two-Step Runge-Kutta pairs

Jackiewicz, Zdzisław; Verner, J. H.

doi:10.1007/bf03167454

Cited by 25 publications

(22 citation statements)

References 24 publications

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“…This suggests that these two problems selected may not adequately challenge the variable step format. Even so, these results show considerably more efficiency for these two problems than corresponding results reported in [13] for two-step Runge-Kutta (TSRK) pairs of the same orders. This suggests that more study of TSRK pairs is needed if such general linear pairs are to compete successfully with RK pairs.…”

Section: Numerical Experimentscontrasting

confidence: 59%

“…The derivation of other special types of general linear methods [1] such as two-step Runge-Kutta pairs [13] remains an area of interest, and some of these may be found that are competitive with currently implemented pairs. Accordingly, the testing provided here forms a basis on which to decide whether such algorithms are likely to lead to more effective codes for solving initial value problems of type (1).…”

Section: Discussionmentioning

confidence: 99%

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Numerically optimal Runge–Kutta pairs with interpolants

Verner

2009

Numer Algor

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Explicit Runge-Kutta pairs are known to provide efficient solutions to initial value differential equations with inexpensive derivative evaluations. Two criteria for selection are proposed with a view to deriving pairs of all orders 6(5) to 9(8) which minimize computation while achieving a user-specified accuracy. Coefficients of improved pairs, their stability regions and coefficients of appended optimal interpolatory Runge-Kutta formulas are provided on the author's website (www.math.sfu.ca/∼jverner). This note reports results of tests on these pairs to illustrate their effectiveness in solving nonstiff initial value problems. These pairs and interpolants may be used for implementation, or else to provide comparison targets for other new types of methods such as explicit general linear methods.

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Section: Numerical Experimentscontrasting

confidence: 59%

Section: Discussionmentioning

confidence: 99%

Numerically optimal Runge–Kutta pairs with interpolants

Verner

2009

Numer Algor

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“…In the Python code, we use the built-in Adams/BDF multistep method with automatic step size control provided by SciPy, which is based on the Netlib VODE implementation [98]. We have performed tests with other Runge-Kutta-based integration schemes [99], but the results did not change, while the runtime of the code increased significantly. This is expected as the ordinary differential equation near the BH becomes stiff, which is a situation not handled well by Runge-Kutta-based integrators.…”

Section: Appendix D: Pyholementioning

confidence: 99%

Chaotic lensing around boson stars and Kerr black holes with scalar hair

et al. 2016

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In a recent paper [P. V. P. Cunha, C. A. R. Herdeiro, E. Radu, and H. F. Runarsson, Phys. Rev. Lett. 115, 211102 (2015).], it was shown that the lensing of light around rotating boson stars and Kerr black holes with scalar hair can exhibit chaotic patterns. Since no separation of variables is known (or expected) for geodesic motion on these backgrounds, we examine the 2D effective potentials for photon trajectories, to obtain a deeper understanding of this phenomenon. We find that the emergence of stable light rings on the background spacetimes allows the formation of "pockets" in one of the effective potentials, for open sets of impact parameters, leading to an effective trapping of some trajectories, dubbed "quasibound orbits." We conclude that pocket formation induces chaotic scattering, although not all chaotic orbits are associated to pockets. These and other features are illustrated in a gallery of examples, obtained with a new ray-tracing code, PYHOLE, which includes tools for a simple, simultaneous visualization of the effective potential, together with the spacetime trajectory, for any given point in a lensing image. An analysis of photon orbits allows us to further establish a positive correlation between photon orbits in chaotic regions and those with more than one turning point in the radial direction; we recall that the latter is not possible around Kerr black holes. Moreover, we observe that the existence of several light rings around a horizon (several fundamental orbits, including a stable one), is a central ingredient for the existence of multiple shadows of a single hairy black hole. We also exhibit the lensing and shadows by Kerr black holes with scalar hair, observed away from the equatorial plane, obtained with PYHOLE.

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“…TSRK methods were introduced by Jackiewicz and Tracogna [18] and further investigated in [1,3,8,9,15,17,20,25], Conte et al (unpublished manuscript) and [26]. Continuous methods (1.1) provide an approximation to the solution y(t) of (1.2) on the whole interval of integration, and not only in the gridpoints {t n } as in the case of discrete TSRK methods (1.3).…”

Section: Introductionmentioning

confidence: 98%