A Fourth Order Multirate Runge-Kutta Method with Error Control

To integrate large systems of ordinary differential equations (ODEs) with disparate timescales, we present a multirate method with error control that is based on embedded, explicit Runge-Kutta (RK) formulas. The order of accuracy of such methods depends on interpolating certain solution components with a polynomial of sufficiently high degree. By analyzing the method applied to a simple test equation, we show that fourth order Cash-Karp RungeKutta coupled with a third order interpolant yields a method that is fourth order overall. We demonstrate our method on three simple test problems and confirm fourth order convergence numerically for each one.