Adapted finite difference schemes advection-reaction-diffusion problems generating periodic wavefronts

The talk is focused on the numerical integration of advection-reaction-diffusion problems by finite difference schemes adapted to problem. In other terms, the numerical scheme, exploiting the a-priori knowledge of the qualitative behaviour of the solution, gains ad- vantages in terms of efficiency and accuracy with respect to classic schemes already known in literature. The adaptation is here carried out through the so-called trigonometrical fitting technique for the discretization in space, giving rise to a system of ODEs whose vector field contains both stiff and non-stiff terms. Due to this mixed nature of the vector field, an Implicit-Explicit (IMEX) method is here employed for the integration in time, based on the first order forward-backward Euler method. The coefficients of the method introduced rely on unknown parameters which have to be properly estimated: such an estimate is performed by minimizing the leading term of the local truncation error in an efficient way. The effectiveness of this problem-oriented approach is shown through a rigorous theoretical analysis and some numerical experiments. References [1] R. D’Ambrosio, M. Moccaldi and B. Paternoster, Adapted numerical schemes for advection-reaction-diffusion problems generating periodic wavefronts, Comp. Math. Appl. (2017). [2] A.J. Perumpanani, J.A. Sherratt, P.K. Maini, Phase differences in reac- tion–diffusion–advection systems and applications to morphogenesis, J. Appl. Math. 55, 19-33 (1995)