Crank-Nicolson Method for the Diffusion Equation | Lecture 72 | Numerical Methods for Engineers

How to construct the Crank-Nicolson method for solving the one-dimensional diffusion equation.
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[@paulegharevba2874]

ONE-DIMENSIONAL DIFFUSION EQUATION SOLUTION METHODS. 1) Explicit Method is CONDITIONALLY stable because with too large time-step, the scheme will blow up. 2) Implicit Method is UNCONDITIONALLY stable irrespective of time-step but every time revolution step requires solving matrix equations 3) Crank-Nicolson Method: is the MOST popular method averaging the explicit and implicit methods; it has the advantage of the implicit and a resulting fast algorithm tri-diagonal matrix equations system (easy to solve) Thank you Sir!

[@zhigangxu2007]

I am interested in seeing you to prove that the Crank-Nicholson method is unconstitutionally stable.

[@user-sx4xj6vz9l]

Thanks for the video, I could solve my problems.

[@NaseerAhmad-dd6pp]

Sir, it will be a long time since any of your videos have been uploaded. I miss you

[@callummurray9100]

thanks for the videos Jeff, great information!

[@RogovAB]

What were the initial and boundary conditions? u=0 at j=1 and 4; and u = 1 at l=0? They were not mentioned in the lecture.

[@gerard-nagle]

Great videos. Can I ask how or what is used to write on the screen?

[@a.vanwijk2268]

2:34 "Both the time stepping and the space stepping become 2nd order". I don't see how the time stepping became 2nd order. In lecture 69 it was still 1st order. Can anyone explain?

[@ProfJeffreyChasnov]

The time stepping becomes a centered difference, which is second order.

[@a.vanwijk2268]

@@ProfJeffreyChasnov So, does this mean then that when one averages the first derivative of two steps, it automatically becomes second derivative?

[@TheYagga]

Same question

[@gowrissshanker9109]

Respected sir , could you please say the geometrical intuition/ interpretation of TRANSPOSE of an MATRIX?

[@skepticmoderate5790]

Transposing a matrix has nothing to do with geometry. It simply means interchanging the rows and columns of a matrix, so that the first row becomes the first column, the second row becomes the second column, etc. For many matrices the transpose is also the inverse, so often taking the transpose of a matrix is a good way to invert it.

[@cleison-em7bk]

Guassian elimination at the end wouldnt be computational expensive ?

[@Jblanco1989]

Yes, it is. In those cases, we can use Crout's algorithm, which is less expensive and converges faster than Gauss's.

[@matesfacilescontesla6034]

Excelent!! How do You write on the board??

[@saurabhtalele1537]

awesome series sir...

[@AJ-et3vf]

Thank you very much sir for this.

[@NaseerAhmad-dd6pp]

Sir Plzzz Upload Videos About Quantum mechanics Mathamatical Model

[@soumyaranimishra245]

Sir, can you please explain the CNM for the 2DBurgers' Equation.

[@NaseerAhmad-dd6pp]

Sir Plzzz can u tell the dimensions of light board u use and the name of mic

[@duncanreeves225]

How do you actually compute the result here? Given that you don't know what U is to start it seems like you have an unknown on both sides of the equation

[@ProfJeffreyChasnov]

Initial condition is given.

[@duncanreeves225]

@@ProfJeffreyChasnov I'm sorry I still don't think I understand how to perform the iterative process. Is there an example somewhere I could see?

[@eylmaz6696]

thanks sir . what is the name of technology you used ? how do you sho yourself like this ?

[@ProfJeffreyChasnov]

It's called a light board.

[@Mau365PP]

Is this in coursera too?

[@ProfJeffreyChasnov]

Yes, sorry. It was private by accident and I just made it public.