# Modeling and Simulation Group Meeting

#### 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.

**Synopsis:**

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) [1] 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

computational physics.

[1] 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.