Modeling and Simulation Group Meeting Old
An adaptive spectral method for oscillatory second-order linear ODEs with frequency-independent cost
Speaker: Fruzsina Agocs, Center for Computational Mathematics, Flatiron Institute
Location: Warren Weaver Hall 1302
Date: Thursday, April 6, 2023, 12:45 p.m.
The efficient numerical solution of highly oscillatory ordinary differential
equations (ODEs) has long been a computational challenge. Conventional methods
require a discretization length proportional to 1/w, where w is
the characteristic frequency, which can get prohibitively slow at large
frequencies. Oscillatory ODEs, however, are ubiquitous in physics and
mathematics, appearing in areas such as cosmology, quantum mechanics, electric
circuitry, plasma physics, gravitational waves, and special function evaluation.
In this talk, I will introduce an efficient numerical method (and its software
implementation)  for second order, homogeneous,
linear ODEs. The solution of the ODE may vary between highly oscillatory and
slowly-changing over the integration range, in which case our solver will
switch between using nonoscillatory (spectral collocation), and a specialized
oscillatory solver to achieve an O(1)
(frequency-independent) runtime. 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.
I will discuss and analyze both underlying methods and how they tie
together algorithmically. I will show how our solver fits into the landscape of
methods for oscillatory ODEs and outperforms other state-of-the-art oscillatory
solvers. Finally, I will illustrate how the solver has been and may be applied in
 Agocs, F. J., & Barnett, A. H. (2022). An adaptive spectral method for oscillatory second-order linear ODEs with frequency-independent cost. arXiv preprint arXiv:2212.06924.