Abstract
In this paper, a new two-parameter family of regularized kernels is introduced, suitable for applying high-order time stepping to N-body systems. These high-order kernels are derived by truncating a Taylor expansion of the non-regularized kernel about \((r^2+\epsilon ^2)\), generating a sequence of increasingly more accurate kernels. This paper proves the validity of this two-parameter family of regularized kernels, constructs error estimates, and illustrates the benefits of using high-order kernels through numerical experiments.
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Notes
If the simplified expression for \(\frac{\partial G^{\epsilon ,n}}{\partial r}\) in Eq. (8) is used, one recovers the same expression involving the hypergeomtric function.
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Acknowledgements
The authors would like to thank Robert Krasny, Keith Cartwright, John Verboncoeur, John Luginsland, Matthew Bettencourt, and Andrew Greenwood for their insightful discussions regarding this work, as well as anonymous referees who have made valuable suggestions to improve the presentation of this manuscript.
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Ong, B.W., Christlieb, A.J. & Quaife, B.D. A New Family of Regularized Kernels for the Harmonic Oscillator. J Sci Comput 71, 1212–1237 (2017). https://doi.org/10.1007/s10915-016-0336-0
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DOI: https://doi.org/10.1007/s10915-016-0336-0