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advection_diffusion_1d.m
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advection_diffusion_1d.m
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% ----------------------------------------------------------------------- %
% __ __ _______ _ ____ _ _ _______ _____ %
% | \/ | /\|__ __| | /\ | _ \| || | /\|__ __| __ \ %
% | \ / | / \ | | | | / \ | |_) | || |_ / \ | | | |__) | %
% | |\/| | / /\ \ | | | | / /\ \ | _ <|__ _/ /\ \ | | | ___/ %
% | | | |/ ____ \| | | |____ / ____ \| |_) | | |/ ____ \| | | | %
% |_| |_/_/ \_|_| |______/_/ \_|____/ |_/_/ \_|_| |_| %
% %
% ----------------------------------------------------------------------- %
% %
% Author: Alberto Cuoci <alberto.cuoci@polimi.it> %
% CRECK Modeling Group <http://creckmodeling.chem.polimi.it> %
% Department of Chemistry, Materials and Chemical Engineering %
% Politecnico di Milano %
% P.zza Leonardo da Vinci 32, 20133 Milano %
% %
% ----------------------------------------------------------------------- %
% %
% This file is part of Matlab4ATP framework. %
% %
% License %
% %
% Copyright(C) 2022 Alberto Cuoci %
% Matlab4ATP is free software: you can redistribute it and/or %
% modify it under the terms of the GNU General Public License as %
% published by the Free Software Foundation, either version 3 of the %
% License, or (at your option) any later version. %
% %
% Matlab4CFDofRF is distributed in the hope that it will be useful, %
% but WITHOUT ANY WARRANTY; without even the implied warranty of %
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the %
% GNU General Public License for more details. %
% %
% You should have received a copy of the GNU General Public License %
% along with Matlab4ATP. If not, see <http://www.gnu.org/licenses/>. %
% %
%-------------------------------------------------------------------------%
% %
% Code: 1D advection-diffusion by the FTCS scheme %
% The code is adapted and extended from Tryggvason, Computational %
% Fluid Dynamics http://www.nd.edu/~gtryggva/CFD-Course/ %
% %
% ----------------------------------------------------------------------- %
close all;
clear variables;
% User-defined data
%-------------------------------------------------------------------------%
np=21; % number of grid points
nstep=100; % number of time steps
L=2.0; % domain length [m]
dt=0.05; % time step [s]
u=1; % velocity [m/s]
D=0.05; % diffusion coefficient [m2/s]
A=0.5; % amplitude of initial solution
k=1; % wave number [1/m]
% Pre-processing of user-defined data
%-------------------------------------------------------------------------%
% Grid step calculation
h=L/(np-1); % grid step [m]
% Memory allocation
fo=zeros(np,1); % temporary numerical solution
f=zeros(np,1); % current numerical solution
a=zeros(np,1); % exact solution
% Initial solution
for i=1:np
f(i)=A*sin(2*pi*k*h*(i-1));
end
% Check the stability conditions on time step
Co = u*dt/h; % Courant number
Di = D*dt/h^2; % Diffusion number
dt_max = min(1*h/u, 0.5*h*h/D); % Maximum allowed time step
fprintf('Co=%f, Di=%f, dt=%f, dt(max)=%f\n', Co, Di, dt, dt_max);
% Video setup
%-------------------------------------------------------------------------%
video_name = 'advection_diffusion_1d.mp4';
videompg4 = VideoWriter(video_name, 'MPEG-4');
open(videompg4);
% Advancing in time
%-------------------------------------------------------------------------%
t = 0.;
for m=1:nstep
% Update the analytical solution
for i=1:np
a(i) = A*exp(-4*pi*pi*k*k*D*t)*sin(2*pi*k*(h*(i-1)-u*t));
end
% Squared areas below the analytical and numerical solutions
a2_int = 0.;
f2_int = 0.;
for i=1:np-1
a2_int = a2_int + h/2*(a(i)^2+a(i+1)^2);
f2_int = f2_int + h/2*(f(i)^2+f(i+1)^2);
end
% Graphical output
message = sprintf('time=%d\na^2(int)=%d\ny^2(int)=%d', t, a2_int, f2_int);
hold off; plot(0:h:L,f,'linewidth',2); axis([0 L -1, 1]); % plot num.
hold on; plot(0:h:L,a,'r','linewidth',2); % plot exact
hold on; legend('numerical', 'exact'); % legend
xlabel('spatial coordinate [m]');
ylabel('solution');
time = annotation('textbox',[0.15 0.8 0.1 0.1],'String',message,'EdgeColor','none');
frame = getframe(gcf);
writeVideo(videompg4,frame);
delete(time);
% Forward Euler method
fo=f;
for i=2:np-1
f(i) = fo(i)-(u*dt/2/h)*(fo(i+1)-fo(i-1))+... % advection
D*(dt/h^2)*(fo(i+1)-2*fo(i)+fo(i-1)); % diffusion
end
% Periodic boundary condition
f(np) = fo(np)-(u*dt/2/h)*(fo(2)-fo(np-1))+...
D*(dt/h^2)*(fo(2)-2*fo(np)+fo(np-1));
f(1) = f(np);
% Update the error between numerical and analytical solution
E = 0;
for i=1:np
E = E + (f(i)-a(i))^2;
end
E = h*sqrt(E);
% New time step
t=t+dt;
% Print the current time (every 25 steps)
if (mod(m,25)==1), fprintf('time=%d E=%e\n', t, E); end
end
% Closing the video stream
close(videompg4);