Numerical treatment of reaction-diffusion problems by trigonometrically-fitted methods

We deal with the numerical solution of partial differential equations, mainly focusing on systems of coupled reaction-diffusion equations, which are known to generate traveling waves as fundamental solutions. Such problems have been typically used as models for life science phenomena exhibiting the generation of periodic waves along their dynamics, e.g. cell cycles, frequently behaving if they are driven by an autonomous biochemical oscillator, or intracellular calcium signalling, since calcium shows many differrent types of oscillations in time and space, in response to various extracellular signals. The periodic character of the problem suggests to propose a numerical solution which takes into account this oscillatory behavior, i.e. by tuning the numerical solver to accurately and efficiently follow the oscillations appearing in the solution, since classical numerical methods would require the employ of a very small stepsize to accurately reproduce the dynamics. For this reason, we propose an adaptation of classical numerical schemes based on finite difference schemes which will take into account the qualitative nature of the solutions. Such schemes provide a twofold level of adaptation to the problem: along space, by means of finite differences based on nonpolynomial fitting techniques, and along time, by means of special purpose numerical time solvers. The coefficients of the resulting methods will depend on the unknown values of parameters related to the problem (e.g. the values of the frequencies of the oscillations): suitable techniques leading to estimates of the parameters will be discussed. Practical constructive aspects and accuracy issues will be treated, as well as numerical experiments showing the effectiveness of the approach will be provided.