Difference methods to one and multidimensional interdiffusion models with Vegard rule

• Published in: Mathematical modelling and analysis Vol. 24; no. 2; pp. 276 - 296
• Main Authors: Bożek, Bogusław, Sapa, Lucjan, Danielewski, Marek
• Format: Journal Article
• Language: English
• Published: Vilnius Vilnius Gediminas Technical University 18.03.2019
• Subjects: Boundary conditions; Computer simulation; Darken method; Differential equations; Diffusion (Physics); Drift; Environmental law; existence and uniqueness of solutions to difference scheme; Existence theorems; Finite difference method; Fluxes; Heat equation; implicit finite difference method; Interdiffusion; Mathematical models; Multidimensional methods; Numerical analysis; parabolic-elliptic nonlinear differential system; Solid solutions; Uniqueness theorems; Vegard rule; interdiffusion; convergence; Darken method; existence and uniqueness of solutions to difference scheme; parabolic-elliptic nonlinear differential system; Vegard rule; implicit finite difference method; stability
• ISSN: 1392-6292; 1648-3510
• DOI: 10.3846/mma.2019.018

In this work we consider the one and multidimensional diffusional transport in an s-component solid solution. The new model is expressed by the nonlinear parabolic-elliptic system of strongly coupled differential equations with the initial and the nonlinear coupled boundary conditions. It is obtained from the local mass conservation law for fluxes which are a sum of the diffusional and Darken drift terms, together with the Vegard rule. The considered boundary conditions allow the physical system to be not only closed but also open. We construct the implicit finite difference methods (FDM) generated by some linearization idea, in the one and two-dimensional cases. The theorems on existence and uniqueness of solutions of the implicit difference schemes, and the theorems concerned convergence and stability are proved. We present the approximate concentrations, drift and its potential for a ternary mixture of nickel, copper and iron. Such difference methods can be also generalized on the three-dimensional case. The agreement between the theoretical results, numerical simulations and experimental data is shown.