Linear Convection Effect of Mesh points

LinearConvection_main.m

clear all
close all
clc

for n=[20, 40, 80, 160] %Looping over List of Grid Points
  %Inputs for linear convection problem
  L = 1;
  c = 1;
  dt = 0.01; %Time Step
  nt = 0:dt:0.4; %No of Time Steps

  x = linspace(0,L,n);
  dx = L /(n-1);

  %start and end of square wave
  x_start = 0.1;
  x_stop = 0.3;

  n_start = nearest_value(x,x_start);
  n_stop  = nearest_value(x,x_stop) ;


  %Initialize arrays and boundary conditions
  u = 0.1 * ones(1, n);
  u(n_start:n_stop) = 0.2;
  uold = u;
  u_initial = u;
  %Time loop
  for k=1:length(nt) %where nt is num of time steps
    %Space loop
    for i=2:n
      u(i) = uold(i) -  c*(dt/dx) * (uold(i) - uold(i-1));
    endfor
    
    %update old velocities
    uold = u;
    
    %Plotting the velocities
    #plot(x, u_initial, \'r\'); 
    #hold on;
    t1 = plot(x, u,\'b\') ;
    xlabel(\'Length of Domain\')
    ylabel(\'Velocity\')
    axis([0 5 0 0.3]) ;
    pause(0.003) 
  endfor
  baseFileName = sprintf(\'%d_GridPoints_LinearConvetion.png\', i);
  legend_string = sprintf(\'%d Grid Points\', i);
  legend(legend_string);
  saveas(t1, baseFileName, \'png\');



endfor

nearest_value.m

function output = nearest_value(x, value_under_test)
  output = find(x - value_under_test > 0 & x - value_under_test < 0.1, 1);
endfunction

Plots:

Explanation:

As seen in the plots the solution is slowly diffusing because of artificial diffusion or false diffusion. So, if we make the mesh finer like in case of n=40 and n=80, the diffusivity decreases and the curve maintains the original velocity profile.
However, the solution blows up after n=100 because of which the plot for n=160 has ringing effects around the points where the function is discontinuous.