On the numerical solution of second order differential equations in the high-frequency regime

Abstract

We describe an algorithm for the numerical solution of second order linear differential equations in the highly-oscillatory regime. It is founded on the recent observation that the solutions of equations of this type can be accurately represented using nonoscillatory phase functions. Unlike standard solvers for ordinary differential equations, the running time of our algorithm is independent of the frequency of oscillation of the solutions. We illustrate the performance of the method with several numerical experiments.

References in corpus (1)

• On the asymptotics of Bessel functions in the Fresnel regime

Cited by in corpus (3)

• An algorithm for the numerical evaluation of the associated Legendre functions that runs in time independent of degree and order
• A quasilinear complexity algorithm for the numerical simulation of scattering from a two-dimensional radially symmetric potential
• On the nonoscillatory phase function for Legendre's differential equation