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Gradient computation

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Gradient Calculation

From Green-Gauss theorem:

\int\limits_V {\nabla \phi dV = } \oint\limits_A {\phi dA}

Written in discrete form:

\left( {\nabla \phi } \right)_0  = {1 \over V}\sum\limits_f {\bar \phi _f A_f }

Value at Face

Cell Based

There are many ways of estimating value of  \phi at face.

  1. Weighted interpolation:  \phi _f  = w\phi _1  + \left( {1 - w} \right)\phi _0
  2. Arithmatic Average:  \phi _f  = 0.5 \left( \phi _1 + \phi _0 \right)

Node Based


The above mentioned gradients are sometimes called unlimited gradients since the face value obtained from them can exceed the bounding cell values. For this reason, for implementing higher order schemes, it becomes important to restrict them, so as not to introduce over and undershoot of variables.

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