Stability Properties of Explicit Runge-Kutta Methods Combined with Richardson Extrapolation

Explicit Runge-Kutta methods of order p with m stages, m = 1, 2, 3, 4, are considered. It is assumed that p = m and that Richardson Extrapolation is additionally used. It is proved that not only are the combinations of the Richardson Extrapolation with the selected explicit Runge-Kutta methods more accurate than the underlying numerical methods, but also their absolute stability regions are considerably larger. Sometimes this fact allows us to apply larger time-stepsizes during the numerical solution when Richardson Extrapolation is used. The possibility to achieve such a positive effect is verified by numerical experiments carried out with a carefully chosen example. It is pointed out that the application of Richardson Extrapolation together with explicit Runge-Kutta methods might be useful when some large-scale mathematical models, including models that are arising in air pollution studies, are handled numerically.