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I am learning Finite Volume method using the textbook "An Introduction to CFD: Finite Volume Method" by Veersteg and Malalasekera.

I would like to solve a heat conduction problem over a square grid $0.5m$ in length with all the boundaries maintained at $0°C$. For that, I need to discretize the Poisson equation:

$$ \frac{d^2 T}{d x^2} + \frac{d^2 T}{d y^2} = 32(x(x-1) + y(y-1)) $$

But I am confused about how to discretize the source term in this since the book does not deal with 2-D non-constant source term. My guess is that we separately integrate each of the terms by $ dxdy$ as below:

$$ \left (A_e \frac{\partial T}{\partial x} - A_w \frac{\partial T}{\partial x}\right) + \left (A_n \frac{\partial T}{\partial y} - A_s \frac{\partial T}{\partial y}\right) = 32x^2dxdy -32xdxdy + 32y^2dxdy - 32ydxdy$$

which becomes:

$$ \left(A_e \frac{T_e - T_p}{\delta x_{ep}} - A_w \frac{T_p - T_w}{\delta x_{pw}} \right) + \left(A_n \frac{T_n - T_p}{\delta y_{np}} - A_s \frac{T_p - T_s}{\delta y_{ps}} \right) = \frac{32}{3}(x^3y + y^3x) - 16(x^2y + y^2x)$$

But I am not sure if this is the right wat to go about it.

I tried to find resources on the internet about this but all I could find was a few academic papers that were not easy to understand.

So what is the correct way to discretize the Poisson equation?

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The correct way is to average the source term which you can do easily in this case as you have a polynomial. In general you can do the average with a quadrature. For second order accuracy if the source term is smooth as in your case, you can also just evaluate it at the cell center, which is like mid-point quadrature. It should not matter much in your example which of these two you do. Try both and see how it affects the solution error.

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  • $\begingroup$ Thanks! By averaging using a quadrature, do you mean performing a numerical integration using Simpson's rule or such? So if I do that i.e. numerically integrate from 0 to 0.5 (the length of the grid), I'll get a number, right? So the source term will become a constant? $\endgroup$ – justauser Mar 3 at 15:27
  • $\begingroup$ Yes, do a numerical quadrature of source term over each cell. Here you can do it exactly since your source term is a polynomial $\endgroup$ – cfdlab Mar 4 at 4:31

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