3d numerical integration over a control volume

hitmre · July 29, 2012, 00:29

I found an equation of numerical integration over of control volume
$\Omega_{ij}=((x,y)|x_{x-1/2}<x<x_{x+1/2} and y_{j-1/2}<y<y_{i+1/2})$
and it says "use nine point stencil" and this equation
$\int _{\Omega_{ij}} g\approx \frac{h^2}{24}(16g_{ij}+g_{i-1,j-1}+g_{i-1,j} +g_{i-1,j+1}+g_{i,j-1}+g_{i,j+1} +g_{i+1,j-1}+g_{i+1,j}+g_{i+1,j+1})$

I am wondering why it is 16/24 and 1/24.
I would like to get the formula for 3D numerical integration, which should based on 3x3x3 points. What the weight should be?

The equation is from this paper (SIAM J. SCI. COMPUT. Vol. 20, No. 4, pp. 1165-1191) on page 1172 http://diyhpl.us/~bryan/papers2/frey...0algorithm.pdf

Thank you.
Jo

vanchanh123 · July 30, 2012, 00:55

Quote:

Originally Posted by hitmre

I found an equation of numerical integration over of control volume
$\Omega_{ij}=((x,y)|x_{x-1/2}<x<x_{x+1/2} and y_{j-1/2}<y<y_{i+1/2})$
and it says "use nine point stencil" and this equation
$\int _{\Omega_{ij}} g\approx \frac{h^2}{24}(16g_{ij}+g_{i-1,j-1}+g_{i-1,j} +g_{i-1,j+1}+g_{i,j-1}+g_{i,j+1} +g_{i+1,j-1}+g_{i+1,j}+g_{i+1,j+1})$

I am wondering why it is 16/24 and 1/24.
I would like to get the formula for 3D numerical integration, which should based on 3x3x3 points. What the weight should be?

The equation is from this paper (SIAM J. SCI. COMPUT. Vol. 20, No. 4, pp. 1165-1191) on page 1172 http://diyhpl.us/~bryan/papers2/frey/distance/Sussman%20M.,%20An%20efficient%20interface%20prese rving%20level%20set%20redistancing%20algorithm.pdf

Thank you.
Jo

In 3D model, I think you can use any the iterpolation way.

The frist question, I think auther use the interpolation method with some weight.

That mean use can choose some coefficentes $a_{i',j',k'}: a_{'i,j',k'}\ge 0, \sum_{(i',j')\in control volume } a_{i,'j',k'}=1$ , when we have interpolation's formular
$\int _{\Omega_{i,j,k}} g =\sum_{(i',j',k')\in control volume} a_{i',j',k'}g_{i',j',k'}.$

July 29, 2012, 00:29	3d numerical integration over a control volume	#1
hitmre New Member mre Join Date: Jul 2012 Posts: 1 Rep Power: 0	I found an equation of numerical integration over of control volume $\Omega_{ij}=((x,y)\|x_{x-1/2}<x<x_{x+1/2} and y_{j-1/2}<y<y_{i+1/2})$ and it says "use nine point stencil" and this equation $\int _{\Omega_{ij}} g\approx \frac{h^2}{24}(16g_{ij}+g_{i-1,j-1}+g_{i-1,j} +g_{i-1,j+1}+g_{i,j-1}+g_{i,j+1} +g_{i+1,j-1}+g_{i+1,j}+g_{i+1,j+1})$ I am wondering why it is 16/24 and 1/24. I would like to get the formula for 3D numerical integration, which should based on 3x3x3 points. What the weight should be? The equation is from this paper (SIAM J. SCI. COMPUT. Vol. 20, No. 4, pp. 1165-1191) on page 1172 http://diyhpl.us/~bryan/papers2/frey...0algorithm.pdf Thank you. Jo

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