OpenFOAM guide/Midpoint rule

The assumption that the value at the centre of a control volume closely represents the mean value throughout the control volume. The midpoint method is second-order accurate. It is given by:

$f \left ( \boldsymbol \Psi \right )_V \approx f \left ( \boldsymbol \Psi_p \right )_{CV}$

Where:

• $\boldsymbol \Psi$ is a quantity;
• p is the centre of the control volume; and
• CV is the control volume.

This assumption leads to:

$\int\limits_V f \left ( \boldsymbol \Psi \right ) dV \approx \boldsymbol \Psi_p V$

Where:

• V is the volume of a control volume.