On the existence of nonoscillatory phase functions for second order differential equations in the high-frequency regime

We observe that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In addition, we describe numerical experiments which illustrate important implications of this fact. For example, that many special functions of great interest --- such as the Bessel functions $J_\nu$ and $Y_\nu$ --- can be evaluated accurately using a number of operations which is $O(1)$ in the order $\nu$. The present paper is devoted to the development of an analytical apparatus. Numerical aspects of this work will be reported at a later date

On the numerical calculation of the roots of special functions satisfying second order ordinary differential equations

We describe a method for calculating the roots of special functions satisfying second order linear ordinary differential equations. It exploits the recent observation that the solutions of a large class of such equations can be represented via nonoscillatory phase functions, even in the high-frequency regime. Our algorithm achieves near machine precision accuracy and the time required to compute one root of a solution is independent of the frequency of oscillations of that solution. Moreover, despite its great generality, our approach is competitive with specialized, state-of-the-art methods for the construction of Gaussian quadrature rules of large orders when it used in such a capacity. The performance of the scheme is illustrated with several numerical experiments and a Fortran implementation of our algorithm is available at the author's website

A frequency-independent solver for systems of first order linear ordinary differential equations

When a system of first order linear ordinary differential equations has eigenvalues of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The cost of representing such solutions using standard techniques typically grows with the magnitudes of the eigenvalues. As a consequence, the running times of standard solvers for ordinary differential equations also grow with the size of these eigenvalues. The solutions of scalar equations with slowly-varying coefficients, however, can be efficiently represented via slowly-varying phase functions, regardless of the magnitudes of the eigenvalues of the corresponding coefficient matrix. Here, we couple an existing solver for scalar equations which exploits this observation with a well-known technique for transforming a system of linear ordinary differential equations into scalar form. The result is a method for solving a large class of systems of linear ordinary differential equations in time independent of the magnitudes of the eigenvalues of their coefficient matrices. We discuss the results of numerical experiments demonstrating the properties of our algorithm.Comment: arXiv admin note: text overlap with arXiv:2308.0328

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

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

The ''phase function'' method to solve second-order asymptotically polynomial differential equations

The Liouville-Green (WKB) asymptotic theory is used along with the Boruvka's transformation theory, to obtain asymptotic approximations of ''phase functions'' for second-order linear differential equations, whose coefficients are asymptotically polynomial. An efficient numerical method to compute zeros of solutions or even the solutions themselves in such highly oscillatory problems is then derived. Numerical examples, where symbolic manipulations are also used, are provided to illustrate the performance of the method