Difference in accuracy between Crank Nicolson and Backward Euler when solving 2D heat equation using finite elements?

I am currently trying to solve a basic 2D heat equation with zero Neumann boundary conditions on a circle. I have compared the results when using Crank Nicolson and Backward Euler and have found that Crank Nicolson does not converge to the exact solution any quicker than when using Backward Euler. I found this quite surprising, but can't find anything in my code that would explain why this is the case.

To try and pinpoint why this is happening I performed the same experiment on a square instead of a circle and found the same results. I then performed the same experiment in 1D as opposed to 2D and again found the same result.

However, when I performed the same experiment using finite differences as opposed to finite elements I did find that using Crank Nicolson my solutions converged to the exact solution far quicker than when using Backward Euler.

If anybody can shed any light on my dilemma I would be very grateful.

Cheers

Answer 1

[Youssef Khmou]

Hi,

Here is a document in which the author makes a comparison between Crank Nicolson, Forward Euler, and Backward Euler in terms of Complexity and Implicitness :

http://www.mathematik.uni-dortmund.de/~kuzmin/cfdintro/lecture8.pdf

[Peter]

Thanks for your help. From the document though I still can't quite decipher why Crank Nicolson would be no better than Backward Euler?