Optimal interpolants for Runge–Kutta ray tracing in inhomogeneous media

Abstract

Runge–Kutta integration schemes are well suited to the determination of ray trajectories in inhomogeneous media. There is a fundamental difference, however, between Runge–Kutta schemes and many other schemes for numerically integrating ordinary differential equations: Runge–Kutta schemes are not based on approximating the continuous trajectory by a polynomial. Consequently, these schemes do not implicitly provide a continuous trajectory; they yield only a series of points through which the ray passes, together with the ray direction at those points. A supplementary method must be devised when a continuous trajectory is required. The accuracy of a continuous trajectory for Runge–Kutta schemes is limited by the error introduced in a single iteration of the integrator. A trajectory that attains this limit is referred to here as optimal. The existing method of calculating trajectories for a widely used Runge–Kutta scheme is, in fact, not optimal. Accordingly, an efficient method of determining optimal intermediate trajectories is presented. This new technique is shown to be superior to the existing method for locating ray–surface intersections and allows accuracy doubling (a recently proposed method for accelerating the analysis of systems with inhomogeneous elements) to be used to full advantage.