On a Robust and Efficient Numerical Scheme for the Simulation of Stationary 3-Component Systems with Non-Negative Species-Concentration with an Application to the Cu Deposition from a Cu-(β-alanine)-Electrolyte

Abstract

Three-component systems of diffusion–reaction equations play a central role in the modelling and simulation of chemical processes in engineering, electro-chemistry, physical chemistry, biology, population dynamics, etc. A major question in the simulation of three-component systems is how to guarantee non-negative species distributions in the model and how to calculate them effectively. Current numerical methods to enforce non-negative species distributions tend to be cost-intensive in terms of computation time and they are not robust for big rate constants of the considered reaction. In this article, a method, as a combination of homotopy methods, modern augmented Lagrangian methods, and adaptive FEMs is outlined to obtain a robust and efficient method to simulate diffusion–reaction models with non-negative concentrations. Although in this paper the convergence analysis is not described rigorously, multiple numerical examples as well as an application to elctro-deposition from an aqueous Cu2+-(β-alanine) electrolyte are presented.

Mathematical Modeling of the Limiting Current Density from Diffusion-Reaction Systems

“…Note that the problem formulation (4a)-(4e) does not include the positivity of the species concentration, which will lead to negative species distributions, as discussed in [14]. A common strategy to include the positivity of the concentrations into a model formulation is the usage of restrained minimization problems, see [17].…”

“…A common strategy to include the positivity of the concentrations into a model formulation is the usage of restrained minimization problems, see [17]. A reformulation of (4a)-(4e) was undertaken in [14] and relies on the introduction of new variables p S = d dx u S , for all S ∈ {M, L, K}. The introduction of the new variables yields the following mathematical problem.…”

“…(i) First note that the systems of ordinary differential Equations (4c)-(4e) and (5c)-(5f) is due to the theorem of Picard-Lindelölf-see [20]-which is uniquely solvable. (ii) A robust and efficient methodology to compute approximations of (7) is described in [14], which is based on a combination of augmented Lagrangian methods, conforming adaptive FEM, quasi-Newtonian methods and homotopy methods. Although the methodology is quite expensive in terms of CPU time, it is efficient in comparison to other state-of-the-art methodologies, cf.…”

“…In Section 2, the mathematical model for the limiting current density will be derived, basing on a three-component system of diffusion-reaction equations describing the species concentrations in the diffusion boundary layer and will be treated with the numerical methodology described in [14]. A method to evaluate the thermodynamic equilibria is stated in [15].…”

“…In Section 3, the model will be applied to a Cu-CO 3 electrolyte and demonstrates the applicability of the methodology, which is described in [14,15], to a real-life problem and will be compared to a well-known model with no reactions in the diffusion boundary layer, described in [13].…”