2.3.9 Update formula in conservation form

After computing e.g. from the fluxes in the cells

$${\color{DEFcolor} f(\rho_i^n)= v_{} \rho_i^n }$$ (97)

the fluxes at cell boundaries

$$f_{i+\frac{1}{2}}^n$$ (98)

that characterize a method, the update can be done by the formula

$$\rho_i^{n+1} = \rho_i^n - \frac{\Delta t}{\Delta x} \left( f_{i+\frac{1}{2}}^n - f_{i-\frac{1}{2}}^n \right) \enspace .$$ (99)

This is the conservation form because the density changes only due to fluxes through the boundaries and is conserved otherwise,

$$\textstyle \sum_{i=i_0}^{i_1} \rho_i^{n+1} \! = \! \textstyle \sum_{i=i_0}^{i_1} \rho_i^{n} + \frac{\Delta t}{\Delta x} \sum_{i=i_0}^{i_1} \left( f_{i+\frac{1}{2}}^n - f_{i-\frac{1}{2}}^n \right)$$ (100)

$$= \! \textstyle \sum_{i=i_0}^{i_1} \rho_i^{n} + \frac{\Delta t}{\Delta... ...{i+\frac{1}{2}}^n - f_{i+\frac{1}{2}}^n \right) - f_{i_0-\frac{1}{2}}^n \right]$$ (101)

$$= \! \textstyle \sum_{i=i_0}^{i_1} \rho_i^{n} + \frac{\Delta t}{\Delta x} \left( f_{i_1+\frac{1}{2}}^n - f_{i_0-\frac{1}{2}}^n \right) \enspace .$$ (102)