A Linearly Fourth Order Multirate Runge-Kutta Method with Error Control

To integrate large systems of locally coupled ordinary differential equations with disparate timescales, we present a multirate method with error control that is based on the Cash-Karp Runge-Kutta formula. The order of multirate methods often depends on interpolating certain solution components with a polynomial of sufficiently high degree. By using cubic interpolants and analyzing the method applied to a simple test equation, we show that our method is fourth order linearly accurate overall. Furthermore, the size of the region of absolute stability is increased when taking many "micro-steps" within a "macro-step." Finally, we demonstrate our method on three simple test problems to confirm fourth order convergence.