% m39_FisherKPP.m - explicit solution of Fisher-KPP equation
%         u_t = eps*u_xx + u - u^2,  u(0)=1, u(10)=0   (nonlinear).

% Grid and initial data:
  eps = 0.01;
  h = .05;                      
  k = .4*h^2/eps;                 % try .4 -> .51
  x = (0+h:h:20-h)';
  u = exp(-2*x) + (abs(x-3)<1);   % initial data: wave front

% Set-up for plot:
  hold off, shg, subplot(2,1,1)
  plt = plot(x,u,'linewidth',4);
  axis([0 20 -.01 1.1]), grid

% Sparse matrix for finite difference operation:
  L = length(x);
  a = (1-2*k*eps/h^2);
  b = k*eps/h^2;
  bc = b*[1; zeros(L-1,1)];
  main = a*sparse(ones(L,1));
  off  = b*sparse(ones(L-1,1));
  A = diag(main) + diag(off,1) + diag(off,-1);
  
% Time-stepping:
  disp('type <return> to see solution')
  pause
  while 1
    u = A*u + bc + k*(u-u.^2);    % Euler's method
    set(plt,'ydata',u)
    drawnow
  end