Adapted numerical methods for advection-reaction-diffusion problems generating periodic wavefronts

We present an adapted method of lines for advection-reaction-diffusion problems generating periodic wavefronts [3], by exploiting the a-priori known information about the qualitative behaviour of the solution. Since the dynamics exhibits a non-polynomial character, classical finite difference methods could require a very small stepsize because they are constructed in order to be exact (within round-off error) on polynomials up to a certain degree. In our approach, the employ of non-polynomially fitted finite differences may guarantee a better balance between accuracy and efficiency requirements. Once a advection-reaction-diffusion problem is discretized in space, the vector field of the resulting system of ordinary differential equations results to be split in two different terms, a stiff term and a nonlinear one. Hence, we propose an implicit-explicit (IMEX) method that implicitly integrates only stiff components and explicitly integrates the nonlinear part, with a significant benefit in terms of efficiency. For the overall numerical scheme, combining the non-polynomial fitting strategy with the IMEX time integration, accuracy and stability properties are rigorously studied, also in comparison with the classical polynomial case [1]. Moreover, since the adapted method has non-constant coefficients depending on unknown parameters linked to the solution, we propose an estimation strategy based on minimization of the leading term of the local discretization error [2]. This is a joint work with Raffaele D'Ambrosio and Beatrice Paternoster (University of Salerno). [1] D'Ambrosio, R., Moccaldi, M., Paternoster, B., Adapted IMEX numerical methods for reaction-diffusion problems, Appl. Numer. Math. (submitted) [2] D'Ambrosio, R., Moccaldi, M., Paternoster, B., Parameter estimation in adapted numerical methods for reaction-diffusion problems, J. Sci. Comput. (submitted) [3] Perumpanani, A.J., Sherratt, J.A., Maini, P.K., Phase differences in reaction-diffusion-advection systems and applications to morphogenesis, IMA J. Appl. Math. 55, 19--33 (1995).