2.3.8 Centering of quantities, fluxes, and differences

Examples for ``natural centering'':

• Quantities at grid points (integer indices): e.g. $\rho_i^n, \rho_{i+1}^n,\ldots$
• Spatial differences of quantities (half-integer $i$ indices): e.g. $\Delta \rho_{i+\frac{1}{2}}^n = \rho_{i+1}^n - \rho_i^n$
• Time differences of quantities (half-integer $n$ indices): e.g. $\Delta \rho_i^{n+\frac{1}{2}} = \rho_i^{n+1} - \rho_i^n$
• Fluxes at half-integer $i$ indices (and - in fact - preferably at half-integer $n$ indices) to get update properly centered: $\Delta \rho_i^{n+\frac{1}{2}} = -\frac{\Delta t}{\Delta x} \left( f_{i+\frac{1}{2}}^n - f_{i-\frac{1}{2}}^n \right)$

Effects of ``natural centering'':

• $\frac{\rho_{i+1}^n - \rho_i^n}{x_{i+1}-x_{i}}$ is $O \! \left( \Delta x\right)$ for $\frac{\partial \rho}{\partial x} \left( x_i \right)$ and $\frac{\partial \rho}{\partial x} \left( x_{i+1} \right)$.
• $\frac{\rho_{i+1}^n - \rho_i^n}{x_{i+1}-x_{i}}$ is $O \! \left( \Delta x^2 \right)$ for $\frac{\partial \rho}{\partial x} \left( x_{i+\frac{1}{2}} \right)$.
• $\frac{\rho_{i+1}^n - \rho_{i-1}^n}{x_{i+1}-x_{i-1}}$ is $O \! \left( \Delta x^2 \right)$ for $\frac{\partial \rho}{\partial x} \left( x_i \right)$.