Crank-Nicolson vs Heun's method


by Smed
Tags: cranknicolson, heun, method
Smed
Smed is offline
#1
Apr6-10, 09:49 AM
P: 28
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%27s_method and for the Crank-Nicolson method I'm looking here http://en.wikipedia.org/wiki/Crank%E...icolson_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[tex]^{n+1}[/tex] = u[tex]^{n}[/tex] - [tex]\frac{1}{2}[/tex]u[tex]^{n}[/tex]dt - [tex]\frac{1}{8}[/tex]u[tex]^{n}[/tex]dt[tex]^{2}[/tex]
Phys.Org News Partner Science news on Phys.org
Review: With Galaxy S5, Samsung proves less can be more
Making graphene in your kitchen
Study casts doubt on climate benefit of biofuels from corn residue
Lord Crc
Lord Crc is offline
#2
Apr9-10, 10:02 AM
P: 88
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
matematikawan is offline
#3
Apr9-10, 10:39 AM
P: 330
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
Lord Crc is offline
#4
Apr10-10, 09:46 AM
P: 88

Crank-Nicolson vs Heun's method


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
Redbelly98 is offline
#5
Apr10-10, 05:57 PM
Mentor
Redbelly98's Avatar
P: 11,988
Quote Quote by Smed View Post
The equation I have for both is:

u[tex]^{n+1}[/tex] = u[tex]^{n}[/tex] - [tex]\frac{1}{2}[/tex]u[tex]^{n}[/tex]dt - [tex]\frac{1}{8}[/tex]u[tex]^{n}[/tex]dt[tex]^{2}[/tex]
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 = ( - un+1 - 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.)


Register to reply

Related Discussions
Crank-Nicolson method for a parabolic differential equation Introductory Physics Homework 8
Approximations used in Crank-Nicolson method for solving PDEs numerically Differential Equations 7
Consistency of Crank Nicolson method Calculus & Beyond Homework 0
Crank-Nicolson method (Matlab) Math & Science Software 0
Parallelizing Crank-Nicolson method Differential Equations 3