Answers - One Dimensional Diffusion Model

      % DIFF1.M    1-Dimensional Diffusion Solution
      %
      % Tom Huber, March 1997
      %
      ncells = 25;       % Number of Cells in the Array
      NTimes = 20;       % Number of Time steps to perform

      FracLeft  = 1/3;   % Fraction diffusing to the left
      FracSame  = 1/3;   % Fraction which remains in same cell
      FracRight = 1/3;   % Fraction diffusing to the right

      Values = zeros(ncells,1);  % Initialize the Array to have zero concentration
      VMax = 100;                % Maximum Concentration
      Values(ncells/2) = VMax;   % Initially set the concentration at the midpoint

      for i=1:NTimes  % Perform a total of NTimes time steps
        NewValues = Values*FracSame;  % New values initially a fraction of original
        for j=2:ncells-1  % For each cell except leftmost or rightmost
              NewValues(j-1) = NewValues(j-1)+Values(j)*FracLeft;  % Diffuse Left
              NewValues(j+1) = NewValues(j+1)+Values(j)*FracRight; % Diffuse Right
        end % for j=1...

        Values = NewValues;     % The updated values become the current values
        plot(Values)            % Plot the values
        axis([1 ncells 0 VMax]) % Set the minima and maxima on axes
        xlabel('Position')      % Label the axes and the title
        ylabel('Value')
        title(['After Time Step: ' num2str(i)])
        drawnow                % Make Matlab display the graph
                         % (Normally it only displays at the end of a program)
      end  % for i=1...

Tutorial Tasks

1. Run the M-file diff1 as written (this performs undriven diffusion, with equal probabilities to the left, right and same cells)
2. Modify the M-file to better model the flow in the river by starting the initial concentration a quarter of the way along the array and changing the fraction to the left to 1/10, the fraction staying in the same location to 3/10, and the fraction to the right to 6/10. Run the program several times with different values for the relative fractions, NMax, NCells, etc.

      % DIFF1.M    1-Dimensional Diffusion Solution
      %
      % Tom Huber, March 1997
      %
      ncells = 25;       % Number of Cells in the Array
      NTimes = 20;       % Number of Time steps to perform

      FracLeft  = 1/10;   % Fraction diffusing to the left
      FracSame  = 3/10;   % Fraction which remains in same cell
      FracRight = 6/10;   % Fraction diffusing to the right

      Values = zeros(ncells,1);  % Initialize the Array to have zero concentration
      VMax = 100;                % Maximum Concentration
      Values(ncells/2) = VMax;   % Initially set the concentration at the midpoint

      for i=1:NTimes  % Perform a total of NTimes time steps
        NewValues = Values*FracSame;  % New values initially a fraction of original
        for j=2:ncells-1  % For each cell except leftmost or rightmost
              NewValues(j-1) = NewValues(j-1)+Values(j)*FracLeft;  % Diffuse Left
              NewValues(j+1) = NewValues(j+1)+Values(j)*FracRight; % Diffuse Right
        end % for j=1...

        Values = NewValues;     % The updated values become the current values
        plot(Values)            % Plot the values
        axis([1 ncells 0 VMax]) % Set the minima and maxima on axes
        xlabel('Position')      % Label the axes and the title
        ylabel('Value')
        title(['After Time Step: ' num2str(i)])
        drawnow                % Make Matlab display the graph
                         % (Normally it only displays at the end of a program)
      end  % for i=1...

3. Start the initial concentration in Cell number 1 - what happens? The model as written has a problem with the leftmost and rightmost points, namely we cannot add onto the points to the left or to the right of the endpoints [which would be NewValues(0) or NewValues(ncells+1) respectively]. We got around this problem by making our loop for j=2:ncells-1, however this causes the program to miss diffusion to the right from the leftmost cell (and to the left from the rightmost cell). Correct this problem by making our loop run from for j=1:ncells and include if statements whereby we have transport to the left if j>1 and transport to the right if j<ncells.

         % DIFF1.M    1-Dimensional Diffusion Solution
         %
         % Tom Huber, March 1997
         %
         ncells = 25;       % Number of Cells in the Array
         NTimes = 20;       % Number of Time steps to perform
   
         FracLeft  = 1/10;   % Fraction diffusing to the left
         FracSame  = 3/10;   % Fraction which remains in same cell
         FracRight = 6/10;   % Fraction diffusing to the right
   
         Values = zeros(ncells,1);  % Initialize the Array to have zero concentration
         VMax = 100;                % Maximum Concentration
         Values(ncells/4) = VMax;   % Initially set the concentration at the midpoint
   
         for i=1:NTimes  % Perform a total of NTimes time steps
           NewValues = Values*FracSame;  % New values initially a fraction of original
           for j=1:ncells  % For each cell except leftmost or rightmost
              if j>1
                 NewValues(j-1) = NewValues(j-1)+Values(j)*FracLeft;  % Diffuse Left
              end
              if j