Differentiation

At $\gcs{1}$ cell faces:

$$\begin{split}\vat{ \dif{q}{\gcs{1}} }{ i + \frac{1}{2} } = \left\{ \begin{alignedat}{2} & \text{Negative wall:} & - \vat{q}{\frac{1}{2}} + \vat{q}{1}, \\ & \text{Positive wall:} & - \vat{q}{\ngp{1}} + \vat{q}{\ngp{1} + \frac{1}{2}}, \\ & \text{Otherwise:} & - \vat{q}{i} + \vat{q}{i + 1}. \end{alignedat} \right.\end{split}$$

At $\gcs{1}$ cell centers:

$$\vat{ \dif{q}{\gcs{1}} }{ i } = - \vat{q}{i - \frac{1}{2}} + \vat{q}{i + \frac{1}{2}}.$$

At $\gcs{2}$ cell faces:

$$\vat{ \dif{q}{\gcs{2}} }{ j + \frac{1}{2} } = - \vat{q}{j} + \vat{q}{j + 1}.$$

At $\gcs{2}$ cell centers:

$$\vat{ \dif{q}{\gcs{2}} }{ j } = - \vat{q}{j - \frac{1}{2}} + \vat{q}{j + \frac{1}{2}}.$$

At $\gcs{3}$ cell faces:

$$\vat{ \dif{q}{\gcs{3}} }{ k + \frac{1}{2} } = - \vat{q}{k} + \vat{q}{k + 1}.$$

At $\gcs{3}$ cell centers:

$$\vat{ \dif{q}{\gcs{3}} }{ k } = - \vat{q}{k - \frac{1}{2}} + \vat{q}{k + \frac{1}{2}}.$$