RK-Slim: Embedded Runge-Kutta without the excess baggage

Abstract

Computationally demanding scientific simulations require numerical integration of large systems of ordinary differential equations (ODEs) in real time. Runge-Kutta (RK) methods with step-size control are frequently employed because they are accurate and have short execution times. Computational step sizes can be increased while maintaining accuracy if the error generated during each time step is below a predetermined threshold. However, if the error exceeds the threshold, the time step must be re-computed using a smaller step-size. In practice, the re-computation is unnecessary for most equations in the system because violations of the error tolerance are localized and occur in just a few equations. We present an efficient and accurate method for solving ODEs that exploits these observations on step-size by eliminating many unnecessary computations in embedded RK methods. We demonstrate how our new method can be a valuable tool for practitioners in the field through simulations with real-world data. © 2011 John Wiley & Sons, Ltd.