Local Error: Euler/Crank-Nicholson

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In summary, Chris seems to think that the Euler and Crank-Nicholson methods are more accurate than the theta reduction method. He doesn't really know how to use the ODE to answer the question, and he is sorry for posting in multiple forums.
  • #1
ChrisHuey
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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
 

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  • #2
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.
 
  • #3
tkhunny said:
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.
 
  • #4
tkhunny said:
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.
 
  • #5
ChrisHuey said:
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.
 

What is a local error in the context of Euler/Crank-Nicholson?

A local error in the context of Euler/Crank-Nicholson is the difference between the exact solution and the numerical approximation at a specific point in the domain. It measures the accuracy of the numerical method at that particular point.

How is the local error calculated in the Euler method?

The local error in the Euler method is calculated by subtracting the exact solution from the numerical approximation at each time step. It is typically denoted as Elocal and can be expressed as Elocal = |y(tn+1) - yn+1|, where y(tn+1) is the exact solution and yn+1 is the numerical approximation at the next time step.

What factors can affect the local error in the Crank-Nicholson method?

The local error in the Crank-Nicholson method can be affected by the step size (h), the time step (Δt), and the order of the method. A smaller step size and time step can result in a smaller local error, while using a higher-order method can reduce the local error as well.

How can the local error be minimized in numerical methods?

The local error in numerical methods can be minimized by using a smaller step size and time step, using a higher-order method, and ensuring that the initial conditions and boundary conditions are accurate. Additionally, performing error analysis and adjusting the parameters accordingly can also help minimize the local error.

What is the significance of the local error in numerical methods?

The local error in numerical methods is important because it allows us to assess the accuracy of our numerical approximations. By analyzing the local error, we can determine the effectiveness and reliability of a particular numerical method and make necessary adjustments to improve its accuracy.

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