Local Error: Euler/Crank-Nicholson

  • Context: MHB 
  • Thread starter Thread starter ChrisHuey
  • Start date Start date
  • Tags Tags
    Error Local
Click For Summary

Discussion Overview

The discussion revolves around the application of numerical methods, specifically the Euler method and the Crank-Nicholson method, in solving an ordinary differential equation (ODE). Participants express uncertainty regarding the application of these methods to a specific problem and seek assistance in understanding the second part of the question related to the ODE.

Discussion Character

  • Homework-related
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant, Chris, notes that substituting theta = 0 yields the Euler method, while theta = 1/2 corresponds to the Crank-Nicholson method, suggesting a difference in accuracy between the two methods.
  • Several participants express frustration over the lack of shown work in the original post, emphasizing the importance of demonstrating one's own efforts in problem-solving.
  • Another participant mentions partial differentiation of the ODE with respect to x and y, providing specific derivatives but expresses uncertainty about their correctness and the overall approach to the problem.
  • There is a recurring theme of participants feeling overwhelmed or unsure about how to proceed with the ODE portion of the question, indicating a struggle with the material.

Areas of Agreement / Disagreement

Participants generally agree on the importance of showing work in problem-solving but express differing levels of understanding regarding the application of the numerical methods and the ODE. The discussion remains unresolved, with multiple competing views on how to approach the problem.

Contextual Notes

Participants mention specific values for theta and their implications for the methods, but there is uncertainty about the correct application of these methods to the ODE. Some assumptions about the problem's requirements and the definitions of variables remain unclear.

ChrisHuey
Messages
4
Reaction score
0
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
 

Attachments

  • NMQ1b.PNG
    NMQ1b.PNG
    10.1 KB · Views: 150
  • ODE.PNG
    ODE.PNG
    405 bytes · Views: 154
Physics news on Phys.org
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.
 
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.
 
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.
 
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.
 

Similar threads

Graduate Crank-Nicholson solution to the cylindrical heat equation
  • · Replies 23 ·
Replies
23
Views
6K
Undergrad Derive local truncation error for the Improved Euler Method
  • · Replies 2 ·
Replies
2
Views
3K
MHB Solving SIR Model with Euler's Method
  • · Replies 7 ·
Replies
7
Views
4K
Graduate Solving Diffusion Problem with Crank Nicholson Method
  • · Replies 1 ·
Replies
1
Views
2K
MHB Euler’s Method y'+2y=2-e^(-4t) y(0)-1
  • · Replies 5 ·
Replies
5
Views
5K
Graduate Heat equation order of accuracy (Crank-Nicolson)
  • · Replies 12 ·
Replies
12
Views
9K
Graduate Problem with Crank-Nicholson for solving hyperbolic equations
  • · Replies 4 ·
Replies
4
Views
5K
MHB How can I apply Euler's method without the interval component?
  • · Replies 8 ·
Replies
8
Views
3K
MATLAB Solving 2nd Order PDE System with Crank-Nicholson
  • · Replies 4 ·
Replies
4
Views
2K
MHB Solving an IVP: Finding Euler Approximations & Errors
  • · Replies 9 ·
Replies
9
Views
4K