Airy Phase Functions

It is well known that phase function methods allow for the numerical solution of a large class of oscillatory second order linear ordinary differential equations in time independent of frequency. Unfortunately, these methods break down in the commonly-occurring case in which the equation has turning points. Here, we resolve this difficulty by introducing a generalized phase function method designed for the case of second order linear ordinary differential equations with turning points. More explicitly, we prove the existence of a slowly-varying ``Airy phase function'' that efficiently represents a basis in the space of solutions of such an equation, and describe a numerical algorithm for calculating this Airy phase function. The running time of our algorithm is independent of the magnitude of the logarithmic derivatives of the equation's solutions, which is a measure of their rate of variation that generalizes the notion of frequency to functions which are rapidly varying but not necessarily oscillatory. Once the Airy phase function has been constructed, any reasonable initial or boundary value problem for the equation can be readily solved and, unlike step methods which output the values of a rapidly-varying solution on a sparse discretization grid that is insufficient for interpolation, the output of our scheme allows for the rapid evaluation of the obtained solution at any point in its domain. We rigorously justify our approach by proving not only the existence of slowly-varying Airy phase functions, but also the convergence of our numerical method. Moreover, we present the results of extensive numerical experiments demonstrating the efficacy of our algorithm.