The method of lines (MOL) for diffusion equations with Neumann boundary conditions is considered. These equations are transformed by a discretization in space variables into systems of ordinary differential equations. The proposed ODEs satisfy the mass conservation law. The stability of solutions of these ODEs with respect to discrete ^{2} norms and discrete

Diffusion is one of several transport phenomena that occur in nature. Many physical processes in metallurgy are controlled by diffusion, for example, oxidation and sintering. There are two ways to introduce the notion of diffusion: an atomistic approach and continuum approach. The central role in the diffusion theory of multicomponent systems plays the Kirkendall effect [

Our ternary system arises from the Fick second law. If the overall molar density ^{3}])

The method proposed by Darken [

We are interested in establishing an approximation method of solutions to the ternary system mentioned above by solutions of associated systems of ordinary differential equations. These systems of ordinary differential equations are obtained by using a discretization in spatial variables.

From the abundant literature concerning the numerical method of lines (MOL) for classical PDEs we mention the monographs [

The aim of the paper is to construct a method of lines for diffusion equations in three-component system with Neumann boundary conditions. We prove stability of approximate solutions with respect to discrete

Our research is concerned with the situation where the initial data and solutions are at least of class

In Section

We consider the system of diffusion equations:

We formulate the method of lines corresponding to (

Denote by

Local existence and uniqueness of solutions of (

Suppose that we have two constants

If the initial data belong to

Suppose that

First, we prove that

We next show that

Denote by

Suppose that

Note that

The stability of the method of lines with respect to discrete

Suppose that

consider

Then we have

Denote by

The proof is based on the following observation:

From Lemma

We define the norm

Suppose that

Then we have

It is easily seen that

Suppose that we have Green functions

Initial values (a) and experimental values of

Evolution of concentrations

Experimental values of

Our numerical experiments confirm the theory and stability analysis. It is seen that the proposed numerical method is stable, mass-conservative, and dissipative. Its solutions do not leave the interval

Experimental values of Lipschitz constants

Experimental values of

We formulate a Gronwall type lemma, which was applied in Section

Suppose that

Since the right-hand sides of system (

If we differentiate these expressions with respect to

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Science Center (Poland) decision no. DEC-2011/02/A/ST8/00280.