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.
Highlights
Runge-Kutta (RK) methods are powerful tools for numerically solving systems of first-order ordinary differential equations, and as such are often the default option in numerical routines for this task
We have presented a numerical solver for second-order ordinary differential equations that can be written in the form of a one-dimensional oscillator, with a time-varying frequency and friction term that do not necessarily have a closed form
We have shown that the solver is significantly more efficient than other known methods if the frequency varies slowly over some part of the integration range, even if it is extremely large, because the solver can exploit the WKB approximation in these cases to traverse many oscillations at once
Summary
Runge-Kutta (RK) methods are powerful tools for numerically solving systems of first-order ordinary differential equations, and as such are often the default option in numerical routines for this task. There are cases when more efficient methods are needed than Runge-Kutta, such as where the solution exhibits rapid oscillations. We develop a method for a more general solution, motivated by the Mukhanov-Sasaki equation [2], which governs the time evolution of curvature perturbations in the early universe. It has the form of a generalized oscillator with a time-dependent frequency and a first-order derivative term present, the frequency depending on the characteristic
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