Finite difference schemes for multilayer diffusion

Abstract

a b s t r a c tAlthough numerical methods have been developed for diffusion through single layer materials, few have been developed for multiple layers. Diffusion processes through a multilayered material are of interest for a wide range of applications, including industrial, biological, electrical, and environmental areas. We present finite difference schemes for multilayered materials with a range of matching conditions between the layers, in particular for a jump matching condition. We show the finite difference methods are flexible, simple to implement, and help illustrate interesting behaviour in multilayered diffusion.

A method of recursive images extended to solve diffusion in multilayer materials under convective boundary conditions

“…(8) employs Laplace transforms so that Eq. (8) appears linked to the inverse Laplace transform of the following expression (see pages 358-9 of the reference for details), 2 ðh=jÞ ffiffiffiffiffiffiffiffi s=D p þ ðh=jÞ Tð2d À xÞ ) 2…”

“…Diffusion is an occurrence which draws interest of a wide range of fields such as of heat, mass, electric charge transport [2,3] and even financial markets. The equation governing this phenomenon and in particular that of diffusion heat, is a partial differential equation [4,5] given in its simplest form by,…”

A method of recursive images to solve transient heat diffusion in multilayer materials

“…Diffusion through multiple layers is an occurrence which has applications in a wide range of areas of heat and mass transport [1,2]. The partial differential equation [3,4] governing this phenomenon and in particular that of the heat diffusion in an N layer material, is given for each layer i in its simplest form by,…”

Critical times in single-layer reaction diffusion

“…Note, we maintain the notation x 0 and x 1 for the boundary positions to better match the notation used in the multilayer diffusion-only analysis in Hickson et al [15,16]. The numerical results presented throughout this article are evaluated using an extension to the finite difference scheme outlined in Hickson et al [15,28]. This code was also verified against a finite elements solution using the commercial package FLEXPDE [29], and against a MATLAB [30] implementation of the exact solution.…”