What's the Difference Between Crank-Nicolson and Heun's Method?

[Smed]

Hi, can someone tell me the difference between the Crank-Nicolson and Heun numerical methods? For Heun's method I'm looking here http://en.wikipedia.org/wiki/Heun's_method and for the Crank-Nicolson method I'm looking here http://en.wikipedia.org/wiki/Crank–Nicolson_method . When I actually carry out a calculation with equal timesteps for both methods and f(t,u)=-.5*u, I get the exact same solution.

The equation I have for both is:

u^{n+1} = u^{n} - \frac{1}{2}u^{n}dt - \frac{1}{8}u^{n}dt^{2}

 

[Lord Crc]

I'm no expert, but from what I can gather Heun's method is for ODE's while Crank-Nicolson is for PDE's?

 

[matematikawan]

Heun's method is an improvement of the forward Euler's method which is an explicit method.
While Crank-Nicolson method is an implicit method. Probably the improvement for the backward Euler method. This is the Crank-Nicolson method for ODE.

But of course the Crank-Nicolson method is verypopular in PDE.

 

[Lord Crc]

Ah, in all cases I've come across Crank-Nicolson, it has been to solve PDEs.

So, if I read my notes correctly, while both methods take an average of the current state and the state at the next timestep, the main difference between Heun's method and Crank-Nicolson is that for Heun's method you use a predictor for the next timestep, keeping it explicit, while for Crank-Nicolson it is used implicitly instead. At least that's my understanding.

Using this I get some different results from yours, both with Crank-Nicolson and Heun's method, are you sure you do Heun's method correctly?

 

[Redbelly98]

The equation I have for both is:

u^{n+1} = u^{n} - \frac{1}{2}u^{n}dt - \frac{1}{8}u^{n}dt^{2}

If we make that a +⅛ instead, I agree that Heun's method gives that equation. But I get something different for Crank-Nicolson.

For Crank-Nicolson, ignore the x-dependence of u and we have

(uⁿ⁺¹ - uⁿ) / Δt = ½ ( -½ uⁿ⁺¹ - ½ uⁿ)​

Solve that for uⁿ⁺¹ and we get something different than the Heun's method equation. (Though they do agree up to order Δt².)