Span-wise operator¶
Average¶
Average at span-wise cell faces:
\[\vat{
\ave{
q
}{
\gz
}
}{
k + \frac{1}{2}
}
=
\frac{1}{2} \vat{q}{k} + \frac{1}{2} \vat{q}{k + 1}\]
Average at span-wise cell centers:
\[\vat{
\ave{
q
}{
\gz
}
}{
k
}
=
\frac{1}{2} \vat{q}{k - \frac{1}{2}} + \frac{1}{2} \vat{q}{k + \frac{1}{2}}\]
Differentiation¶
Differentiation at span-wise cell faces:
\[\vat{
\dif{
q
}{
\gz
}
}{
k + \frac{1}{2}
}
=
-
\vat{q}{k}
+
\vat{q}{k + 1}\]
Differentiation at span-wise cell centers:
\[\vat{
\dif{
q
}{
\gz
}
}{
k
}
=
-
\vat{q}{k - \frac{1}{2}}
+
\vat{q}{k + \frac{1}{2}}\]
Summation¶
Summation of a quantity defined at span-wise cell faces:
\[\sumzf q
\equiv
\vat{q}{\frac{1}{2}}
+
\vat{q}{\frac{3}{2}}
+
\cdots
+
\vat{q}{\nz - \frac{3}{2}}
+
\vat{q}{\nz - \frac{1}{2}}\]
Summation of a quantity defined at span-wise cell centers:
\[\sumzc q
\equiv
\vat{q}{1}
+
\vat{q}{2}
+
\cdots
+
\vat{q}{\nz - 1}
+
\vat{q}{\nz}\]