\underConstruction
The background to the SIMPLE, PISO and PIMPLE pressure-velocity algorithms can be demonstrated using the incompressible, inviscid flow equations, comprising the momentum equation:
\[\ddt{\u} + \div (\u \otimes \u) = - \grad p,\] \[\div \u = 0.\]Discretising the momentum equation leads to a set of algebraic equations of the form:
\[M[\u] = - \grad p\]where the matrix \(M[\u]\) comprises the diagonal and off-diagonal contributions using the decomposition:
\[M[\u] = A\u - \vec{H}\]The discretised momentum equation therefore becomes:
\[A\u - \vec{H} = - \grad p\]which on re-arranging leads to the velocity correction equation:
\[\u = \frac{\vec{H}}{A} - \frac{1}{A} \grad p.\]The volumetric flux corrector equation is then derived by interpolating \(\u\) to the faces and dotting the result with the face area vectors, \(\vec{S}_f\):
\[\phi = \u_f \dprod \vec{S}_f = \left( \frac{\vec{H}}{A} \right)_f \dprod \vec{S}_f - \left( \frac{1}{A} \right)_f \vec{S}_f \dprod \snGrad p\]Discretisation of the continuity equation yields the constraint:
\[\div \phi = 0.\]Substituting the flux equation leads to the pressure equation:
\[\div \left[ \left( \frac{1}{A} \right)_f \grad p \right] = \div \left( \frac{\vec{H}}{A} \right)_f.\]