# An efficient method for solving highly oscillatory ordinary differential equations with applications to physical systems

## TL;DR

This paper introduces oscode, a novel numerical method combining WKB approximation and adaptive Runge-Kutta techniques to efficiently solve highly oscillatory ODEs, with applications in physics and cosmology.

## Contribution

The paper presents oscode, a new computational routine that improves efficiency in solving oscillatory differential equations by adaptively switching between WKB and Runge-Kutta methods.

## Key findings

- Successfully solves Airy and oscillatory equations with high accuracy.
- Efficiently computes quantum and cosmological perturbations.
- Outperforms existing codes like BINGO in speed and accuracy.

## Abstract

We present a novel numerical routine (oscode) with a C++ and Python interface for the efficient solution of one-dimensional, second-order, ordinary differential equations with rapidly oscillating solutions. The method is based on a Runge-Kutta-like stepping procedure that makes use of the Wentzel-Kramers-Brillouin (WKB) approximation to skip regions of integration where the characteristic frequency varies slowly. In regions where this is not the case, the method is able to switch to a made-to-measure Runge-Kutta integrator that minimises the total number of function evaluations. We demonstrate the effectiveness of the method with example solutions of the Airy equation and an equation exhibiting a burst of oscillations, discussing the error properties of the method in detail. We then show the method applied to physical systems. First, the one-dimensional, time-independent Schr\"odinger equation is solved as part of a shooting method to search for the energy eigenvalues for a potential with quartic anharmonicity. Then, the method is used to solve the Mukhanov-Sasaki equation describing the evolution of cosmological perturbations, and the primordial power spectrum of the perturbations is computed in different cosmological scenarios. We compare the performance of our solver in calculating a primordial power spectrum of scalar perturbations to that of BINGO, an efficient code specifically designed for such applications.