Untitled

The following simple ODE uncovers a surprise.

f = @(t,u) u.^2 - u.^3;
u0 = 0.005;

We will solve the problem first with the implicit AM2 method using

steps.

[tI,uI] = am2(f,[0 400],u0,200);

plot(tI,uI)

xlabel('t'), ylabel('u(t)') % ignore this line

title('AM2 solution') % ignore this line

Now we repeat the process using the explicit AB4 method.

[tE,uE] = ab4(f,[0 400],u0,200);

hold on, plot(tE,uE), ylim([-1 2])

title('AM2 and AB4 solutions') % ignore this line

Once the solution starts to take off, the AB4 result goes catastrophically wrong.

format short e, uE(105:111)
ans = 
   7.5539e-01
   1.4373e+00
  -3.2890e+00
   2.1418e+02
  -4.4821e+07
   4.1269e+23
  -3.2214e+71

We hope that AB4 will converge in the limit

, so let's try using more steps.

for n = [1000 1600]

[tE,uE] = ab4(f,[0 400],u0,n);

plot(tE,uE)

end

So AB4, which is supposed to be more accurate than AM2, actually needs something like 8 times as many steps to get a reasonable-looking answer!