AverageΒΆ
At \(\gcs{1}\) cell faces:
\[\begin{split}\vat{
\ave{q}{\gcs{1}}
}{
i + \frac{1}{2}
}
=
\left\{
\begin{alignedat}{2}
& \text{Negative wall:} & \vat{q}{\frac{1}{2}}, \\
& \text{Positive wall:} & \vat{q}{\ngp{1} + \frac{1}{2}}, \\
& \text{Otherwise:} & \frac{1}{2} \vat{q}{i} + \frac{1}{2} \vat{q}{i + 1}.
\end{alignedat}
\right.\end{split}\]
At \(\gcs{1}\) cell centers:
\[\vat{
\ave{q}{\gcs{1}}
}{
i
}
=
\frac{1}{2} \vat{q}{i - \frac{1}{2}}
+
\frac{1}{2} \vat{q}{i + \frac{1}{2}}.\]
At \(\gcs{2}\) cell faces:
\[\vat{
\ave{q}{\gcs{2}}
}{
j + \frac{1}{2}
}
=
\frac{1}{2} \vat{q}{j}
+
\frac{1}{2} \vat{q}{j + 1}.\]
At \(\gcs{2}\) cell centers:
\[\vat{
\ave{q}{\gcs{2}}
}{
j
}
=
\frac{1}{2} \vat{q}{j - \frac{1}{2}}
+
\frac{1}{2} \vat{q}{j + \frac{1}{2}}.\]
At \(\gcs{3}\) cell faces:
\[\vat{
\ave{q}{\gcs{3}}
}{
k + \frac{1}{2}
}
=
\frac{1}{2} \vat{q}{k}
+
\frac{1}{2} \vat{q}{k + 1}.\]
At \(\gcs{3}\) cell centers:
\[\vat{
\ave{q}{\gcs{3}}
}{
k
}
=
\frac{1}{2} \vat{q}{k - \frac{1}{2}}
+
\frac{1}{2} \vat{q}{k + \frac{1}{2}}.\]