Numerical solution of partial differential equations by IMEX methods based on non-polynomial fitting

This talk deals with the numerical solution of partial differential equations, with special attention to problems that, after a spatial semi-discretization of the operator, reduce to a system of ODEs whose vector field can be split into two terms: one generating stiff components in the solution and one giving rise to nonstiff ones. Such problems can be efficiently treated by suitable IMEX numerical methods, as clearly visible from the existing literature. We will mainly focus our attention on systems of partial differential equations represented in terms of two coupled reaction-diffusion equations profitably solvable by IMEX methods and which are known to generate traveling waves as fundamental solutions [4,5]. 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 [3], frequently behaving if they are driven by an autonomous biochemical oscillator; intracellular calcium signalling [1,5], 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 IMEX schemes based on finite differences which will take into account the qualitative nature of the solutions. Extending the ideas in [2], we may say that a three-fold level of adaptation to problem will be carried out: along time and space, by suitable semidiscretization with problem-based finite differences and analog time solvers for the semi-discrete problem, and along the problem by taking into account the peculiarity of the vector field through the employ of IMEX schemes. The approximant will be constructed in order to exactly integrate (within round-off error) problems whose solution lies in a finite dimensional linear space (the so-called fitting space) spanned by a set of functions other than polynomials, properly chosen to achieve the desired level of tuning to the problem. The corresponding numerical method will depend on variable coefficients, which are functions of the parameters characterizing the solution (e.g. the frequency of the oscillations). Thus, we handle two main aspects: (i) choosing a fitting space which is as much as possible suitable to represent the solution of the problem; (ii) accurately computing/estimating the parameters on which the numerical method depends. We show how these aspects can be accurately approached by taking into account the existing theoretical studies on the problem. Practical constructive aspects and accuracy issues will be treated, as well as numerical experiments showing the effectiveness of the approach will be provided. References [1] M.J. Berridge, Calcium oscillations, J. Biol. Chem. 265, 9583–9586 (1990). [2] R. D’Ambrosio, B. Paternoster, Numerical solution of a diffusion problem by exponentially fitted finite difference methods, Springer Plus 3, 425–431 (2014). [3] J.E. Ferrell, T.Y. Tsai, Q. Yang, Modeling the cell cycle: why do certain circuits oscillate?, Cell. 144(6), 874885 (2011). [4] N. Kopell, L.N. Howard, Plane wave solutions to reaction-diffusion equations, Stud. Appl. Math. 52, 291328 (1973). [5] J.A. Sherratt, Periodic waves in reaction-diffusion models of oscillatory biological systems, FORMA 11, 6180 (1996).