It's just a generalization of CN with effect of:
* having L-diagonal matrices (rather than 3-diagonal for CN)
* having s time-steps iterations (rather than 1 for CN)
The later form they put as repeated applications of the LU, so you get psi_{n+i/s} for i =1, ..., s, as intermediate steps...
Okay but then I am completely confused with respect to the time gain with LU.
By using the highest Padé expansion & five time steps method from this text [http://goo.gl/nLB6X] , I find, for a 2^8 real-space grid:
Computation of the propagator: 0.26s
Calculation of 2^9 time steps: 0.1s...
Dear AlephZero and Hurkyl,
> [...] 1-dimensional heat transfer/diffusion finite difference scheme using Crank Nicholson. To get rid of any complications because of boundary conditions, assume the temperature of both end points are prescribed functions of time.
Okay, the general idea is now...
maybe by this you mean inversion of a matrix propagates numerical error in a way LU doesn't? But if this so, why isn't it documented? Why is the fact not spelt out to say that LU is required, rather than matrix inversion (and thus time is not the relevant parameter but some numerical stability...
You're right that LU decomposition is faster than matrix inversion, but as it has to be done once only, and takes seconds on larger spaces than use in practise, the gain is not compelling, given that it's much easier to write the code that gives psi_{n+1} in a single computation of C psi_n than...
Dear all,
Thanks for your replies.
To Hootenanny & Hurkyl, are you saying it's not equivalent to solve A psi_{n+1} = B psi_n and psi_{n+1} = A^{-1} B psi_n ? (I've never met the case where A is singular). If so, in which sense is it not? I compared both and I see no discrepancy in...
When solving differential equations numerically with finite difference methods, textbooks get to the point of solving:
A psi_(n+1) = B psi_n
(with A, B some matrices, typically complex conjugate of each other) and advise on using LU decomposition to do so.
My question is, why not...