Studying absolute stability properties of the Richardson Extrapolation combined with explicit Runge–Kutta methods

Abstract

Explicit Runge–Kutta methods are considered. It is assumed that the number of stages m,m=1,2,3,4, is equal to the order p of the selected method. The impact of the application of the Richardson Extrapolation on the absolute stability properties is studied. The Richardson Extrapolation was used until now only in an attempt to increase the accuracy of the numerical approximations or in order to keep the computational errors under some prescribed in advance level. Another issue, the absolute stability of the Richardson Extrapolation in connection with several numerical methods, is the major topic of this study. It is shown that not only are the combinations of the Richardson Extrapolation with explicit Runge–Kutta methods more accurate than the underlying numerical methods, but also their absolute stability regions are larger. This means that larger time-stepsizes can be used during the integration when Richardson Extrapolation is used. The validity of the theoretical results is confirmed by numerical experiments with three carefully chosen examples. 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, described by systems of partial differential equations, are handled numerically.