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Finite Difference Implicit Method for Fick's 2nd Law

version 1.0.0 (1.97 KB) by Roche de Guzman

Roche de Guzman (view profile)

PDE solution using the implicit method

Updated 26 Apr 2019

%% Finite Difference Implicit Method (Matrix Equations) 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
% Matrix of coefficients
muD = (dt/dx^2)*D;
M = diag([1 ones(1,nx-2)*(1+2*muD) -1])... % main diagonal
+ diag([-1 ones(1,nx-2)*-muD],1)... % 1st superdiagonal
+ diag([ones(1,nx-2)*-muD 1],-1); % 1st subdiagonal
% Vector of unknowns per time point
for j = 1:nt-1
K = [0; c(2:nx-1,j); 0]; % column vector of constants
U = M\K; % column vector of unknowns
c(:,j+1) = U; % concentration matrix
end

Cite As

Roche de Guzman (2019). Finite Difference Implicit Method for Fick's 2nd Law (https://www.mathworks.com/matlabcentral/fileexchange/71358-finite-difference-implicit-method-for-fick-s-2nd-law), MATLAB Central File Exchange. Retrieved .

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MATLAB Release Compatibility
Created with R2019a
Compatible with any release
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