Efficient implementation of stable Richardson Extrapolation algorithms

Richardson Extrapolation is a powerful computational tool which can successfully be used in the efforts to improve the accuracy of the approximate solutions of systems of ordinary differential equations (ODEs) obtained by different numerical methods (including here combined numerical methods consisting of appropriately chosen splitting procedures and classical numerical methods). Some stability results related to two implementations of the Richardson Extrapolation (Active Richardson Extrapolation and Passive Richardson Extrapolation) are formulated and proved in this paper. An advanced atmospheric chemistry scheme, which is commonly used in many well-known operational environmental models, is applied in a long sequence of experiments in order to demonstrate the fact that (a) it is indeed possible to improve the accuracy of the numerical results when the Richardson Extrapolation is used (also when very difficult, badly scaled and stiff non-linear systems of ODEs are to be treated), (b) the computations can become unstable when the combination of the Trapezoidal Rule and the Active Richardson Extrapolation is used, (c) the application of the Active Richardson Extrapolation with the Backward Euler Formula is leading to a stable computational process, (d) experiments with different algorithms for solving linear systems of algebraic equations are very useful in the efforts to select the most suitable approach for the particular problems solved and (e) the computational cost of the Richardson Extrapolation is much less than that of the underlying numerical method when a prescribed accuracy has to be achieved.