File:Midpoint method illustration.png

 

% illustration of numerical integration
% compare the Forward Euler method, which is globally O(h) 
% with Midpoint method, which is globally O(h^2)
% and the exact solution

function main()

   f = inline ('2-y', 't', 'y');    % will solve y' = f(t, y)

   a=0; b=1; % endpoints of the interval where we will solve the ODE
   A = -0.5*b; B = 1.5*b; % a bit of an expanded interval
   N = 2; T = linspace(a, b, N); h = T(2)-T(1); % the grid
   y0 = 1; % initial condition

%   % One step of the midpoint method
   Y = solve_ODE (N, f, y0,  h, T, 2); % midpoint method

   % exact solution to the right  
   hh=0.05; TT = a:hh:B; NN = length(TT);
   YY = solve_ODE (NN, f, y0,  hh, TT, 2); % midpoint method

   % exact solution to the left 
   TTl = a:hh:(-A); NN = length(TTl);
   ZZ = solve_ODE (NN, f, y0,  -hh, TTl, 2); % midpoint method

%  the tangent line at the midpoint
   tmid = (a+b)/2;
   I = find(TT >= tmid); m = I(1);
   tmid = TT(m); ymid = YY(m); slope = f(tmid, ymid);
   Tan_l = 0.5*b; Tant = (tmid-Tan_l):hh:(tmid+Tan_l); Tany = slope*(Tant-tmid)+ymid; 

%  prepare the plotting window
   lw = 3; % curves linewidth
   lw_thin  = 2; % thinner curves
   fs = 30; % font size
   figure(1); clf; hold on; axis equal; axis off;

% colors
   red=[0.867 0.06 0.14];
   blue = [0, 129, 205]/256;
   green = [0, 200,  70]/256;
   black = [0, 0, 0];

% coordinate axes
   shifty=0.2;
   arrowsize=0.1; arrow_type=1; angle=20; % in degrees
   arrow([A, shifty], [B, shifty], lw_thin, arrowsize, angle, arrow_type, black)

% plot auxiliary lines
   I = find(TT >= a); m = I(1);  ya = YY(m);
   plot([a, a], [0+shifty, ya], 'linewidth', lw_thin, 'linestyle', '--', 'color', black)

   I = find(TT >= tmid); m = I(1);  ymid = YY(m);
   plot([tmid, tmid], [0+shifty, ymid], 'linewidth', lw_thin, 'linestyle', '--', 'color', black)

   I = find(TT >= b); m = I(1);  yb = YY(m);
   plot([b, b], [0+shifty, yb], 'linewidth', lw_thin, 'linestyle', '--', 'color', black)

% plot the solutions
   plot(TT, YY, 'color', blue,   'linewidth', lw);
   plot(-TTl, ZZ, 'color', blue,   'linewidth', lw)
   plot(T, Y, 'color', red, 'linewidth', lw)

   % plot the tangent line
   plot(Tant, Tany+0.003*lw, 'color', green, 'linewidth', lw)

   smallrad = 0.02;
   ball (T(1), Y(1), smallrad, red)
   ball (T(length(T)), Y(length(Y)), smallrad, red)
   
% text
   small = 0.15; 
   text(a, shifty-small, '\it{t_n}', 'fontsize', fs)
   text(tmid, shifty-small, '\it{t_n+h/2}', 'fontsize', fs)
   text(b, shifty-small, '\it{t_{n+1}}', 'fontsize', fs)
   text(T(1)-1.5*small, Y(1), '\it{y_n}', 'fontsize', fs, 'color', red)
   text(T(length(T))+0.6*small, Y(length(Y)), '\it{y_{n+1}}', 'fontsize', fs, 'color', red)
   text(-TTl(length(TTl))+0.1*small, ZZ(length(ZZ))+3*small, '\it{y(t)}', 'fontsize', fs, 'color', blue)
   
   
   % axes aspect ratio
%   pbaspect([1 1.5 1]);

%% save to disk
   saveas(gcf, sprintf('Midpoint_method_illustration.eps', h), 'psc2');
   
function Y = solve_ODE (N, f, y0,  h, T, method)

   Y = 0*T;
   
   Y(1)=y0;
   for i=1:(N-1)
	  t = T(i); y = Y(i);

	  if method == 1 % forward Euler method
		 
		 Y(i+1) = y + h*f(t, y);
		 
	  elseif method == 2 % explicit one step midpoint method
		 
		 K = y + 0.5*h*f(t, y);
		 Y(i+1) =  y + h*f(t+h/2, K);
		 
	  else
		 disp ('Don`t know this type of method');
		 return;
		 
	  end
   end


   function arrow(start, stop, thickness, arrow_size, sharpness, arrow_type, color)

% Function arguments:
% start, stop:  start and end coordinates of arrow, vectors of size 2
% thickness:    thickness of arrow stick
% arrow_size:   the size of the two sides of the angle in this picture ->
% sharpness:    angle between the arrow stick and arrow side, in degrees
% arrow_type:   1 for filled arrow, otherwise the arrow will be just two segments
% color:        arrow color, a vector of length three with values in [0, 1]

% convert to complex numbers
   i=sqrt(-1);
   start=start(1)+i*start(2); stop=stop(1)+i*stop(2);
   rotate_angle=exp(i*pi*sharpness/180);

% points making up the arrow tip (besides the "stop" point)
   point1 = stop - (arrow_size*rotate_angle)*(stop-start)/abs(stop-start);
   point2 = stop - (arrow_size/rotate_angle)*(stop-start)/abs(stop-start);

   if arrow_type==1 % filled arrow

      % plot the stick, but not till the end, looks bad
      t=0.5*arrow_size*cos(pi*sharpness/180)/abs(stop-start); stop1=t*start+(1-t)*stop;
      plot(real([start, stop1]), imag([start, stop1]), 'LineWidth', thickness, 'Color', color);

      % fill the arrow
      H=fill(real([stop, point1, point2]), imag([stop, point1, point2]), color);
      set(H, 'EdgeColor', 'none')

   else % two-segment arrow
      plot(real([start, stop]), imag([start, stop]),   'LineWidth', thickness, 'Color', color);
      plot(real([stop, point1]), imag([stop, point1]), 'LineWidth', thickness, 'Color', color);
      plot(real([stop, point2]), imag([stop, point2]), 'LineWidth', thickness, 'Color', color);
   end

function ball(x, y, r, color)
   Theta=0:0.1:2*pi;
   X=r*cos(Theta)+x;
   Y=r*sin(Theta)+y;
   H=fill(X, Y, color);
   set(H, 'EdgeColor', 'none');