Numerical methods for spatio-temporal oscillating solutions in reaction-diffusion models

This talk concerns numerical approximations of oscillating patterns in reaction-diffusion systems. Stability analysis on a test reactiondiffusion system with oscillating solution allows comparisons in terms of stability regions and stepsize restrictions of some time integrators. Moreover, a dissipation and dispersion analysis are also carried out, see [1]. We consider the well known and commonly used IMEX Euler method, the explicit and semi-implicit ADI methods and two new methods, symplectic in absence of diffusion, named IMSP and IMSP IE. For the discretization in the space Extended Central Difference Formulas (ECDFs) of order p=2,4,6 (ECDF p) are applied. The theoretical results are supported by numerical experiments. Firstly we consider the space dependent Lotka-Volterra model, with spatially homogeneous solution oscillating only in time, then we solve the Schnackenberg model, prototype of reaction-diffusion system with Turing-Hopf patterns oscillating both in space and time. Moreover, we present numerical simulations of Turing-Hopf patterns for the morphochemical model for metal growth described in[3]. We point out that both IMSP and IMSP IE methods are more efficient for the LotkaVolterra model in presence of diffusion. Nevertheless, in the case of Turing patterns oscillating in space and time, the explicit ADI method reaches good accuracy with lower computational cost.