An accelerated frequency-independent solver for oscillatory differential equations

Abstract Oscillatory differential equations arise in many numerical and scientific calculations. Because the running times of standard solvers for ordinary differential equations (ODEs) increase linearly with frequency when applied to such problems, a variety of specialized methods, most of them quite complicated, have been proposed. Here we point out that one of the simplest conceivable approaches not only works, but yields a scheme for solving oscillatory second-order linear ordinary differential equations which is significantly faster than current state-of-the-art techniques. Our method, which operates by constructing a slowly-varying phase function representing a basis in the space of solutions of the differential equation, runs in time independent of the frequency of oscillations of the solutions and can be applied to second-order equations whose solutions are oscillatory in some regions and slowly varying in others. In the high-frequency regime, our algorithm discretizes the nonlinear Riccati equation satisfied by the derivative of the phase function via a Chebyshev spectral collocation method and applies the Newton–Kantorovich method to the resulting system of nonlinear algebraic equations. We prove that the iterates converge quadratically to a nonoscillatory solution of the Riccati equation. The quadratic convergence of the Newton–Kantorovich method and the simple form of the linearized equations ensure that this procedure is extremely efficient. Our algorithm then extends the slowly-varying phase function calculated in the high-frequency regime throughout the solution domain by solving a certain third-order linear ordinary differential equation related to the Riccati equation. Once the slowly-varying phase function has been constructed, any reasonable initial or boundary value problem can be readily solved and its solution can be evaluated anywhere in the differential equation’s domain at a cost which is independent of frequency. We describe the results of numerical experiments demonstrating the properties of our scheme and comparing it with state-of-the-art methods for the solution of oscillatory differential equations.