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3d numerical integration over a control volume

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

Quote:

Originally Posted by hitmre (Post 374148)

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
$\int _{\Omega_{i,j,k}} g =\sum_{(i',j',k')\in control volume} a_{i',j',k'}g_{i',j',k'}.$