Errors obtained in the numerical solution of the 1D convection-diffusion equation

I have been trying to solve the 1D convection-diffusion equation

This is my code so far:

clc; clear; close all

%Input parameters
Ai     = 0; %Initial concentration in the column
Ao    = 1;      %Influent concentration
L     = 0.08;       %Length of column: m
nx    = 40;           %number of spatial gridpoints (i.e., no. of cells)
dx    = L/nx;           %Length step size [m]
T     = 20/24;      %End time [days]... that is 20hrs
nt    = 100;            %shifts
dt    = T/nt;       %Time step size [days] (Always check stability!)
Vel   = dx/dt;      %Velocity in each cell [m/day]
alpha = 0.002;      %Dispersivity [m]
De    = alpha*Vel;   % Dispersion coeff. [m2/day]

%Gridblocks
x    = 0:dx:L;
t    = 0:dt:T;

%Initial and boundary conditions
g1     = @(t) Ao;
g2     = @(t) 0;
A       =  zeros(nx+1, nt+1);
A(:,1) =  Ai;
A(1,:) =  g1(t);

% Dimensionless parameters
beta      = Vel*dt/dx;
gamma = De*dt/(dx^2);

% Implementation of the explicit method
for j= 1:nt-1      % Time Loop
for i= 2:nx-1  % Space Loop
A(i,j+1) = (A(i-1,j))*(beta + gamma)...
+ A(i,j)*(1-2*gamma-beta) + A(i+1,j)*(gamma);
end

% Insert boundary conditions for i = 1 and i = N
A(2,j+1) =  A(1,j)*(beta + gamma) + A(2,j)*(1-2*gamma-beta) + A(3,j)*(gamma);
A(nx,j+1) =  A(nx-1,j)*(beta + 2*gamma) + A(nx,j)*(1-2*gamma-Vel*beta)+ (2*gamma*dx*g2(t));
end

figure
plot(t, A(end,:), 'r*', 'MarkerSize', 2)
title('A Vs time profile (Using FDM)')
xlabel('t'),ylabel('A')


When I run the code, I obtain negative values of the dependent variable, however, comparison with MATLAB’s pdepe (see plot) gives a different profile, which I believe is the correct (matches with the analytical solution) result.

I am not sure what exactly I could be doing wrong here, although it seems that my major shortcoming is in the implementation of the initial and boundary conditions. I am thus hoping that someone may be able to help me out or point me in the right direction. Thank you in advance.

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– Wasabi
Jul 1, 2021 at 20:40