== The Finite Difference Technique ==
The general form of Poisson's equation for any potential ψ temperature can be expressed as:
<math> \frac{\partial^2\psiT}{\partial x^2} + \frac{\partial^2\psiT}{\partial y^2} + \frac{\partial^2\psiT}{\partial z^2} = -f(\mathbf{r}) </math>
When f(<b>r</b>) = 0, one obtains the well-known Laplace equation, which applies to source-free regions.
The second derivative of ψ T with respect to the x coordinate can be approximated by the second-order difference:
<math> \frac{\partial^2\psiT(\mathbf{r})}{\partial x^2} \approx \frac{\psiT(x+\Delta x,y,z)-2\psi2T(x,y,z)+\psiT(x-\Delta x,y,z)}{(\Delta x)^2} </math>
Similar expressions can be written for the second derivative with respect to the y and z coordinates.
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The potential twmperature at the point (x,y,z) can be expressed in terms of the potentials temperatures at its six neighboring grid points along the principal axes. This creates a 7-point computational molecule shown in the figure below:
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In the special case of a uniform grid with Δx = Δy = Δz, it can be shown that in a source-free region:
<math> \psiT(i,j,k) = \frac{1}{6} \big[ \psiT(i+1,j,k) + \psiT(i-1,j,k) + \psiT(i,j+1,k) + \psiT(i,j-1,k) + \psiT(i,j,k+1) + \psiT(i,j,k-1) \big] </math>
Two types of domain boundary conditions can be applied: