Multiple Richardson Extrapolation Applied to Explicit Runge–Kutta Methods

Richardson extrapolation has long been used to enhance the accuracy of time integration methods for solving differential equations. The original version of Richardson extrapolation is based on a suitable linear combination of numerical solutions obtained by the same numerical method with two different time-step sizes. This procedure can be extended to more than two step sizes in a natural way, and the resulting method is called repeated Richardson extrapolation. In this talk we investigate another possible generalization of the idea of Richardson extrapolation, where the extrapolation is applied to the combination of some underlying method and the classical Richardson extrapolation. The procedure obtained in this way, called multiple Richardson extrapolation, is analysed for accuracy and absolute stability when combined with some explicit Runge–Kutta methods.