Summation

At \(\gcs{1}\) cell faces:

\[\sumxf q \equiv \vat{q}{\frac{1}{2}} + \vat{q}{\frac{3}{2}} + \cdots + \vat{q}{\ngp{1} - \frac{1}{2}} + \vat{q}{\ngp{1} + \frac{1}{2}}.\]

At \(\gcs{1}\) cell centers:

\[\sumxc q \equiv \vat{q}{1} + \vat{q}{2} + \cdots + \vat{q}{\ngp{1} - 1} + \vat{q}{\ngp{1}}.\]

At \(\gcs{2}\) cell faces:

\[\sumyf q \equiv \vat{q}{\frac{1}{2}} + \vat{q}{\frac{3}{2}} + \cdots + \vat{q}{\ngp{2} - \frac{3}{2}} + \vat{q}{\ngp{2} - \frac{1}{2}}.\]

At \(\gcs{2}\) cell centers:

\[\sumyc q \equiv \vat{q}{1} + \vat{q}{2} + \cdots + \vat{q}{\ngp{2} - 1} + \vat{q}{\ngp{2}}.\]

At \(\gcs{3}\) cell faces:

\[\sumzf q \equiv \vat{q}{\frac{1}{2}} + \vat{q}{\frac{3}{2}} + \cdots + \vat{q}{\ngp{3} - \frac{3}{2}} + \vat{q}{\ngp{3} - \frac{1}{2}}.\]

At \(\gcs{3}\) cell centers:

\[\sumzc q \equiv \vat{q}{1} + \vat{q}{2} + \cdots + \vat{q}{\ngp{3} - 1} + \vat{q}{\ngp{3}}.\]