MHB Local Error: Euler/Crank-Nicholson

[ChrisHuey]

Hi All,

For the attached question, I think that substituting theta = 0, I get the Euler method back. If I substitute theta = 1/2, I get the Crank-Nicholson (Modified Euler) method back.

In terms of accuracy, I know this means that Crank-Nicholson is the more accurate method.

I am mostly unsure how to use the ODE to answer the second part of the question. Not really sure what is being asked.

Is anyone able to help with this? It seems a niche, as I can find little on this particular bit in any notes. Took me long enough to even realize I could reduce it using those values for theta.

Thanks.

Chris

 

[tkhunny]

Posting the same question - with no work shown - n multiple websites is bad form. This says that your time is more valuable than our volunteers. That's no good. Please show YOUR work. You can even take ideas from other answers you may have received.

 

[ChrisHuey]

Posting the same question - with no work shown - n multiple websites is bad form. This says that your time is more valuable than our volunteers. That's no good. Please show YOUR work. You can even take ideas from other answers you may have received.

The theta reduction to Euler and Crank-Nicholson is all I know how to do for the problem. I'm an engineering student who knows how to follow method for this particular topic, and I struggle with it enough as it is.

I have no idea how to do the ODE bit basically, or even what Xn / Xn+1 is for this question to start with.

My apologies for posting in multiple forums with this question, but it is the last problem I have for this, and I have honestly tried hard to get to even where I am with it.

 

[ChrisHuey]

Posting the same question - with no work shown - n multiple websites is bad form. This says that your time is more valuable than our volunteers. That's no good. Please show YOUR work. You can even take ideas from other answers you may have received.

You said partial differentiation. If I partially differentiate the ODE with respect to x, I get 1. With respect to y gives -2y.

It's not that I don't want to show working, but I know the above isn't correct, and I don't actually know what I need to do to answer the question. It's not that I am not willing to try. Just stupid in this respect to this.

 

[tkhunny]

You said partial differentiation. If I partially differentiate the ODE with respect to x, I get 1. With respect to y gives -2y.

It's not that I don't want to show working, but I know the above isn't correct, and I don't actually know what I need to do to answer the question. It's not that I am not willing to try. Just stupid in this respect to this.

Well, it appears you'll need the second and third, not just the first.