Numerical Analysis of Interdiffusion in Multi-Component Systems

Finite element method is used to obtain numerical solution to the Danielewski-Holly model of interdiffusion for non-ideal, multi-component open systems and concentration dependent diffusivities. The definition of the week solution of the initial-boundary-value problem of interdiffusion is formulated. Using the Galerkin scheme, the approximate solution to the problem has been derived. The numerical solution is used for calculations of the components’ concentration profiles in a diffusion couple and compared with the analytical results showing excellent agreement. Introduction In modern mathematics and its applications in science we increasingly more often resort to generalized definition of a solution when the phenomena are described by the partial differential equations, PDEs. Here by generalized (or weak) solution we understand a function that satisfies some integral equation which is equivalent to the original differential equation. However, this equivalence is limited only to some regular class of functions. There are several reasons why weak solutions are used: 1. The existence of a solution is easier to obtain. 2. Extended class of physical phenomena can be described, especially those where noncontinuous or non-differentiable functions are involved. 3. Effective numerical schemes are available. Some of them (e.g. finite element methods) are the most powerful tools now available in applied mathematics. It should be emphasized that in some situations the use of a weak solution is not a mater of mere convenience but of necessity. There are phenomena where functions which describe the real physical quantities are not continuous or differentiable (e.g. shock waves). In this paper we applied the idea of generalized solutions to the problem of interdiffusion in multi-component open systems. A derivation of the mathematical model has been already formulated [1] which after rearrangements leads to the following classical initial-boundary value PDE problem [2]