A step-size selection strategy for explicit Runge–Kutta methods based on Lyapunov exponent theory

Using a stability result for variable time-step Runge–Kutta methods applied to nonautonomous real linear scalar test problem that decays exponentially fast, a step-size selection algorithm is devised. The step-size selection algorithm is based partly on stability information obtained by estimating the discrete Lyapunov exponent of a Runge–Kutta method applied to a nonautonomous linear scalar problem. The utility of the approach is illustrated in numerical experiments that demonstrate how the new algorithm performs against a standard accuracy based step-size selection strategy.