This work concerns the numerical solution of λ-ω reaction-diffusion systems by using a problemoriented approach. Since these systems have travelling waves as fundamental solutions , they are largely used to model life science phenomena that are characterized by the generation of periodic waves along their dynamics. For example, they describe cell cycles that are driven by an autonomous biochemical oscillator  and intracellular calcium signaling, since the concentration of calcium oscillate in time and space according to various extracellular signals. The goal of this work is developing a new numerical method by using information we have about the nature of the problem and the expression of the exact solution. Indeed, Kopell and Howard in  have proved that the considered problem has an one-parameter family of periodic wave solutions. We have applied the method of lines to the problem and we have approximated the spatial second derivatives with adaptedd finite differences extending the ideas in . Such formulas have been constructed in order to be exact (within round-off error) on functions belonging to a finite-dimensional space (called fitting space). For this technique, two main problems deserve careful treatment: the choice of a proper fitting space and the estimate of the parameters in the variable coefficients of the fitted formulas. The system of ODEs arising from spatial semi-discretization is characterized by a stiff component and a non-linear one. So it has been solved by using an IMEX method, that implicitly integrate the first term and explicitly integrate the second one. This approach allows to reach stability without increasing too much the computational cost. We will show the theoretical study of the properties of the new method and the results of numerical experiments. References  J.A. Sherratt, On the evolution of periodic plane waves in reaction-diffusion systems of λ-ω type, SIAM J. Appl. Math. 54, 1374-1385 (1994).  J.E. Ferrell, T.Y. Tsai, Q. Yang, Modeling the cell cycle: why do certain circuits oscillate?, Cell. 144(6), 874885 (2011).  A. Atri, J. Amudson, D. Clapham, J. Sneyd, A single-pool model for intracellular calcium oscillations and waves in Xenopus laevis Oocyte, Biophysical Journal 65, 1727-1739 (1993).  N. Kopell, L.N. Howard, Plane waves solutions to reaction-diffusion equations, Studies in Applied Mathematics 52, 291--328 (1973).  R. D'Ambrosio, B. Paternoster, Numerical solution of reaction-diffusion systems of λ-ω type by trigonometrically fitted methods, J. Comput. Appl. Math., in press.
|Titolo:||Implicit - explicit (IMEX) methods for reaction-diffusion systems with non-polynomial fitting|
|Data di pubblicazione:||2015|
|Appare nelle tipologie:||4.2 Abstract|