Poisson PDE discretisation help

In summary, you can evaluate f(x,y) at each center point i,j of your 5 point finite difference scheme. You can use the values in the cells in the diagram, or the boundary conditions for u.
  • #1
maistral
240
17
Okay, I'm trying to play around again :D

A little overview; I know that the Poisson equation is supposed to be:
uxx + uyy = f(x,y)

I can manage to discretise the partial derivative terms by Taylor. I don't know how to deal with the f(x,y) though. Say for example, uxx + uyy = -exp(x). what values of x will I use?

If possible, by virtue of this Laplace equation solution diagram,

https://fbcdn-sphotos-b-a.akamaihd.net/hphotos-ak-prn2/q71/1533880_710799648952993_1272413896_n.jpg

which values of x will I use, is it the 30's or the 100's? If I add y in f(x,y) as well (perhaps changing the equation to -exp(x+y) as an example), which values of y should I use? is it the 50's or the 100s at the right side? Thanks a lot. :D
 
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  • #2
You evaluate f(x,y) at each center point i,j of your 5 point finite difference scheme.

Chet
 
  • #3
Chestermiller said:
You evaluate f(x,y) at each center point i,j of your 5 point finite difference scheme.

Chet

If I'm correct (hopefully) you meant that I should evaluate f(x,y) given the (x,y) values of the point in the grid, yes?

But which x and y values should I use; for x in the diagram there's 30 and 100, for y there's 50 and 100? :|

https://fbcdn-sphotos-d-a.akamaihd.net/hphotos-ak-prn2/t1/1551523_711126242253667_2047676360_n.jpg

Or am I doing it incorrectly? :eek:
 
Last edited:
  • #4
maistral said:
If I'm correct (hopefully) you meant that I should evaluate f(x,y) given the (x,y) values of the point in the grid, yes?

But which x and y values should I use; for x in the diagram there's 30 and 100, for y there's 50 and 100? :|

https://fbcdn-sphotos-d-a.akamaihd.net/hphotos-ak-prn2/t1/1551523_711126242253667_2047676360_n.jpg

Or am I doing it incorrectly? :eek:

I don't know what the values in the cells in the diagram are, but I suspect they are values of [itex]u[/itex]. If so, the outermost cells do not give [itex]x[/itex] and [itex]y[/itex] values but the boundary conditions for [itex]u[/itex].
 
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  • #5
pasmith said:
I don't know what the values in the cells in the diagram are, but I suspect they are values of [itex]u[/itex]. If so, the outermost cells do not give [itex]x[/itex] and [itex]y[/itex] values but the boundary conditions for [itex]u[/itex].
I agree.

Chet
 
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  • #6
So I'm doing it incorrectly. How do I compute for it?
 
  • #7
Ah wait, nevermind. I think I remembered something. Thanks a lot.
 
  • #8
Lol yay alright! I got it. I can't believe I actually forgot something that basic. *ashamed*

Thanks a lot again :D

https://fbcdn-sphotos-f-a.akamaihd.net/hphotos-ak-frc3/t1/q81/s720x720/1538901_711533678879590_913247690_n.jpg
 

Related to Poisson PDE discretisation help

1. What is a Poisson PDE discretisation?

A Poisson PDE discretisation is a numerical method used to approximate the solution of a Poisson partial differential equation (PDE). It involves dividing the continuous PDE into smaller, discrete parts and solving for the unknown values at each point. This allows for easier computation and analysis of the PDE.

2. Why is discretisation necessary for solving Poisson PDEs?

Discretisation is necessary because it allows for the use of numerical methods to approximate the solution of a PDE. Solving a PDE analytically is often not possible, so discretisation provides a way to obtain an approximate solution that can still be useful for practical purposes.

3. What are the different types of Poisson PDE discretisation methods?

There are several types of discretisation methods for solving Poisson PDEs, including finite difference, finite element, and spectral methods. Each method has its own advantages and disadvantages, and the choice of method depends on the specific problem at hand.

4. How accurate are Poisson PDE discretisation methods?

The accuracy of a discretisation method depends on various factors such as the type of method used, the size of the discretisation grid, and the complexity of the PDE. In general, these methods can provide reasonably accurate solutions, but the accuracy can be improved by using smaller grid sizes and more advanced methods.

5. What are some common challenges when using Poisson PDE discretisation methods?

One common challenge is choosing an appropriate discretisation method for a specific problem. Another challenge is determining the appropriate grid size, as a finer grid may provide a more accurate solution but also require more computational resources. Another challenge is dealing with boundary conditions and how they affect the discretisation process.

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