In this paper we study the numerical approximation of Turing patterns corresponding to steady state solutions of a PDE system of reaction–diffusion equations modeling an electrodeposition process. We apply the Method of Lines (MOL) and describe the semidiscretization by high order finite differences in space given by the Extended Central Difference Formulas (ECDFs) that approximate Neumann boundary conditions (BCs) with the same accuracy. We introduce a test equation to describe the interplay between the diffusion and the reaction time scales. We present a stability analysis of a selection of timeintegrators (IMEX 2-SBDF method, Crank–Nicolson (CN), Alternating Direction Implicit (ADI) method) for the test equation as well as for the Schnakenberg model, prototype of nonlinear reaction–diffusion systems with Turing patterns. Eventually, we apply the ADIECDF schemes to solve the electrodeposition model until the stationary patterns (spots & worms and only spots) are reached.Wevalidate the model by comparison with experiments on Cu film growth by electrodeposition.

### Numerical approximation of Turing patterns in electrodeposition by ADI methods

#### Abstract

In this paper we study the numerical approximation of Turing patterns corresponding to steady state solutions of a PDE system of reaction–diffusion equations modeling an electrodeposition process. We apply the Method of Lines (MOL) and describe the semidiscretization by high order finite differences in space given by the Extended Central Difference Formulas (ECDFs) that approximate Neumann boundary conditions (BCs) with the same accuracy. We introduce a test equation to describe the interplay between the diffusion and the reaction time scales. We present a stability analysis of a selection of timeintegrators (IMEX 2-SBDF method, Crank–Nicolson (CN), Alternating Direction Implicit (ADI) method) for the test equation as well as for the Schnakenberg model, prototype of nonlinear reaction–diffusion systems with Turing patterns. Eventually, we apply the ADIECDF schemes to solve the electrodeposition model until the stationary patterns (spots & worms and only spots) are reached.Wevalidate the model by comparison with experiments on Cu film growth by electrodeposition.
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11580/20650`
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