OpenFOAM guide/Midpoint rule

From OpenFOAMWiki

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}


  • \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


  • V is the volume of a control volume.