Implementation of convection scheme given by normalized variable diagram

Question

In finite difference and finite volume methods, convection schemes (upwind, central, quick, ...) are usually shown in a normalized variable diagram. The diagram gives the normalized face variable as a function of the normalized upwind or downwind variable.

For example, let $D$ be the donor cell, $A$ the acceptor cell and $U$ the upwind cell, then the normalized variable $\tilde{\phi}_f$ at the face between $D$ and $A$ is given as a function of the normalized donor variable $\tilde{\phi}_D$: \begin{equation} \tilde{\phi}_f = F(\tilde{\phi}_D) \quad \text{with} \quad \tilde{\phi}\equiv\frac{\phi-\phi_U}{\phi_A - \phi_U} \end{equation} Various limiters are also expressed in terms of the normalized variables.

How to define and robustly implement such schemes when $\phi_A \approx \phi_U$? (which probably happens somewhere in the flow field).

Answer 1

In the mean time, I've found an answer on pages 24--25 of this lecture. Translated to the notation above, Eq.(15) from page 24 states \begin{equation} \tilde{\phi} = \frac{\phi-\phi_U}{\phi_A - \phi_U} = 1 - \frac{\phi_A-\phi}{\phi_A-\phi_U} \end{equation} This expression is then overruled by Eq.(16) from page 25 such that \begin{equation} \tilde{\phi}=0.5 \quad \text{ if $|\phi_A-\phi| \leq 10^{-6}$ or $|\phi_A - \phi_U| \leq 10^{-6}$} \end{equation} This seems to imply that the central scheme is being used whenever the variation in $\phi$ is small, because $\tilde{\phi}=0.5 \Leftrightarrow \phi=\frac{1}{2}(\phi_U+\phi_A)$.