SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Numerical approximation of Turing patterns in a reaction-diffusion model for electrodeposition

I. Sgura, D. Lacitignola and B. Bozzini

Abstract
We describe the dynamics of metal growth by electrodeposition by a reaction-diffusion PDE system that allows spatial pattern formation through Turing instability [1]. Oscillating Turing patterns arise if a forcing frequency is applied [2]. Numerical approximation of Turing patterns is a challenging task: high accuracy in space and long-time integration are needed, in the forced model highly oscillatory solutions are expected. We perform space semi-discretization by high order finite difference ECDFs. For time discretization, we introduce a test equation and define its stability region in terms of reaction and diffusion time scales. We present a stability analysis for Crank-Nicolson, IMEX 2-SBDF, ADI schemes. We approximate both stationary and oscillating Turing patterns by ADI-ECDF since do not require stepsize restrictions [3, 4]. New results will be presented related to Turing-Hopf instability, which can yield patterns oscillatory in space and time.

Bibliography
[1] B. Bozzini, D. Lacitignola, C. Mele and I. Sgura, Coupling of Morphology and Chemistry Leads to Morphogenesis in Electrochemical Metal Growth: a Review of the Reaction-Diffusion Approach, Acta Appl. Math. 122 (1) (2012) 53-68
[2] B. Bozzini, D. Lacitignola and I. Sgura, Frequency as the greenest additive for metal plating I. J. Elec. Sc 6 (2011) 4553-4571
[3] I. Sgura, B. Bozzini and D. Lacitignola, Numerical approximation of Turing patterns in electrodeposition by ADI methods J. Comput. Appl. Math. 236 (2012) 4132-4147
[4] I. Sgura, B. Bozzini and D. Lacitignola, Numerical approximation of oscillating Turing patterns AIP Conf. Proc 1493 (2012) 896-903

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