Finite Difference Explicit Method for Fick's 2nd Law

Finite Difference Explicit Method (iteration) to solve for a PDE (Fick's 2nd Law of diffusion)
451 Downloads
Updated Fri, 26 Apr 2019 05:30:48 +0000

View License

%% Finite Difference Explicit Method (Iteration) with D = diffusivity: Fick's 2nd Law of Diffusion
% by Prof. Roche C. de Guzman
clear; clc; close('all');
%% Given
xi = 0; xf = 0.6; dx = 0.04; % x range and step size = dx [m]
xL = 0; xU = 0.1; % initial value x lower and upper limits [m]
ti = 0; tf = 0.05; dt = 4e-4; % t range and step size = dt [s]
ci = 2; % initial concentration value [ng/L]
cLU = 8; % initial concentration value within x lower and upper limits [ng/L]
D = 1.5; % diffusivity or diffusion coefficient [m^2/s]
%% Calculations
% Independent variables: x and t
X = xi:dx:xf; nx = numel(X); T = ti:dt:tf; nt = numel(T); % x and t vectors and their number of elements
[x,t] = meshgrid(X,T); x = x'; t = t'; % x and t matrices
% Dependent variable: c
c = ones(nx,nt)*ci; % temporary c(x,t) matrix with rows: c(x) and columns: c(t)
% Initial values and Dirichlet boundary
I = find((X>=xL)&(X<=xU)); % index of lower and upper limits
c(I,1) = cLU; % c at t = 0 for lower and upper limits
% Iteration using Taylor's Finite Difference
for j = 1:nt-1 % t counter
for i = 1:nx-2 % x counter
c(i+1,j+1) = c(i+1,j)+((dt/(dx)^2)*D*(c(i+2,j)-2*c(i+1,j)+c(i,j))); % c(x+dx,t+dt)
% Neumann boundary (zero flux): cx(xi,t)=0 and cx(xf,t)=0 -> c(xi,t)=c(xi+dx,t) and c(xf,t)=c(xf-dx,t)
c(1,j+1) = c(2,j+1); % change c(xi) = c(xi+dx)
c(nx,j+1) = c(nx-1,j+1); % change c(xf) = c(xf-dx)
end
end

Cite As

Roche de Guzman (2024). Finite Difference Explicit Method for Fick's 2nd Law (https://www.mathworks.com/matlabcentral/fileexchange/71361-finite-difference-explicit-method-for-fick-s-2nd-law), MATLAB Central File Exchange. Retrieved .

MATLAB Release Compatibility
Created with R2019a
Compatible with any release
Platform Compatibility
Windows macOS Linux

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!
Version Published Release Notes
1.0.0