% Final, problem 1: Trapezoidally time-differenced FEM for
%
%        dq/dt + dq/dx = 0, 0<x<1, 0<t
%        q(x,0) = 0, q(0,t) = sin(50t).
%
% on uniform and nonuniform grid. Solution is plotted at time
% T=0.9. Uses function finalp1_FEMsolve.m

  dt = 0.005;
  T = 0.9;

  % Part (a):  Solve on a uniform grid dx = 0.01.

  x = 0.01*(0:100); 
  q = finalp1_FEMsolve(x,dt,T);

  t0 = T-x;
  t0(t0<0)=0;
  qex = sin(50*t0);
  plot(x,qex,'r--',x,[qex(1) q'],'b-');
  xlabel('x')
  ylabel('q(x,t=0.9)')

  % Part (b): Now consider nonuniform grid. 
  
  X = 0.02*(0:50);
  x = 1 - (1 - X).^0.5;
  q = finalp1_FEMsolve(x,dt,T);
  hold on
  plot(x,[qex(1) q'],'g-.')
  legend('Exact','FEM(unif dx)','FEM(var dx)')
  hold off