On the employ of time series in the numerical treatment of differential equations modelling oscillatory phenomena

We present a numerical scheme to integrate systems of differential equations modelling phenomena characterized by an oscillatory dynamics. In particular, we numerically treat problems for which time series of experimental data are available in order to exploit all the deriving a-priori known information. We focus on what is now considered the prototype oscillator, the Belousov-Zhabotinsky reaction, which can be modelled by a system of kinetics equations for the concentrations of the key elements. We numerically integrate this system of differential equations (commonly known as Oregonator) following the strategyof exponential fitting [3]. Indeed, classical methods could require a very small step-size to accurately reproduce the oscillatory behaviour of the exact solution because they are developed in order to be exact (within round-off error) on polynomials up to a certain degree. We rather propose a method that is constructed in order to be exact on functions other than polynomials. These basis functions are supposed to belong to a finite dimensional space (the so-called fitting space) and are properly chosen according to the behaviour of the exact solution. As a consequence, the coefficients of the resulting adapted method are no longer constant as in the classical case, but rely on a parameter linked to exact solution, whose value is clearly unknown. Therefore, we need to choose a proper fitting space and estimate the above-mentioned parameter. We show how dealing with these aspects by taking into account the existing theoretical studies on the problem and extracting useful information from the time series of experimental data. Indeed, we select a trigonometrical fitting space because of the a-priori known oscillatory dynamics occurring in BZ reaction. In this case, the basis functions depend on a parameter representing the frequency of the oscillations of the exact solution. We propose estimating this parameter by minimizing the leading term of the local truncation error [2]. However, since the time series of experimental data are available [5], we can select the frequency of the observed oscillations as an approximation of the parameter, thus avoiding expensive procedures involving the resolution of non-linear systems of equations as in [1]. Numerical experiments will be provided to show the effectiveness of the presented approach. To summarize, trigonometrically fitted methods may guarantee a better balance between accuracy and efficiency than classical ones in case of problems having an oscillatory dynamics. However, they require the computation of the parameter which can be very expensive. This limit is overtaken when time series of experimental data are given because the parameter can be estimated without any increase in the computational cost.