An Adaptive Spectral Method for Oscillatory Second-Order Linear ODEs with Frequency-Independent Cost
Abstract.
We introduce an efficient numerical method for second-order linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. In oscillatory regions the solution is generated via a nonoscillatory phase function that obeys the nonlinear Riccati equation. We propose a defect correction iteration that gives an asymptotic series for such a phase function; this is numerically approximated on a Chebyshev grid with a small number of nodes. For analytic coefficients we prove that each iteration, up to a certain maximum number, reduces the residual by a factor of order of the local frequency. The algorithm adapts both the stepsize and the choice of method, switching to a conventional spectral collocation method away from oscillatory regions. In numerical experiments we find that our proposal outperforms other state-of-the-art oscillatory solvers, most significantly at low to intermediate frequencies and at low tolerances, where it may use up to \(10^6\) times fewer function evaluations. Even in high-frequency regimes, our implementation is on average 10 times faster than other specialized solvers.
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Acknowledgments.
We have benefited greatly from discussions with Jim Bremer, Charlie Epstein, Manas Rachh, and Leslie Greengard. The input of the anonymous referees also led to several improvements. The Flatiron Institute is a division of the Simons Foundation.
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Submitted: 20 January 2023
Accepted: 26 September 2023
Published online: 29 January 2024
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Funding: The work of the authors was supported by the Flatiron Institute, a division of the Simons Foundation.
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