Absolute Stability and Implementation of the Two-Times Repeated Richardson Extrapolation Together with Explicit Runge-Kutta Methods

Abstract

Efficient implementation of the Two-times Repeated Richardson Extrapolation is studied in this paper under the assumption that systems of ordinary differential equations (ODEs) are solved numerically by Explicit Runge-Kutta Methods (ERKMs). The combinations of the Two-times Repeated Richardson Extrapolation with the ERKMs are new numerical methods. The computational cost per step of these new numerical methods is higher than the computational cost per step of the underlying ERKMs. However, the order of accuracy of the combined methods becomes very high: if the order of accuracy of the underlying ERKM is p, then the order of accuracy of its combination with the Two-times Repeated Richardson Extrapolation is at least $p+3$ when the right-hand-side function of the system of ODEs is sufficiently many times continuously differentiable. Moreover, the stability properties of the new methods are always better than those of the underlying numerical methods when $p=m$ and $m=1,2,3,4$ (where m is the number of stage vectors in the chosen ERKM). These two useful properties, higher accuracy and better stability, are often giving a very reasonable compensation for the increased computational cost per step, because the same degree of accuracy can be achieved by applying a large stepsize which leads to a considerable reduction of the number of steps when the Two-times Repeated Richardson Extrapolation is used. This fact is verified by several numerical experiments.