Stability of the Richardson Extrapolation combined with some implicit Runge-Kutta methods

The implementation of the Richardson Extrapolation in combination with different numerical methods for solving systems of ordinary differential equations (ODEs) is relatively simple, but the important requirement for stability of the computational process may cause serious difficulties. For example, the commonly used by scientists and engineers Trapezoidal Rule has good stability properties, but its combination with the Richardson Extrapolation is unstable. Therefore, it is necessary to study in advance and very carefully the stability of the new numerical methods arising when the scientists and the engineers use this computational device in combination with different algorithms for solving systems of ODEs.We are presenting a systematic investigation of the implementation of Richardson Extrapolation for two implicit Runge-Kutta methods. Three numerical examples, including an atmospheric chemical scheme used successfully in several extensive environmental studies and described mathematically by a very stiff and badly scaled nonlinear system of ODEs, are presented to illustrate the advantages of the presented approach. The numerical results show that not only are the computations stable, but also the achieved accuracy is higher when the Richardson Extrapolation is additionally applied. It will be possible to derive similar stability and accuracy results for other implicit Runge-Kutta methods.