2.3.16 Derivations of donor cell scheme

Donor cell scheme (FTBS, stencil in Sect. 2.3.11, example later in Sect. 2.4.5):

$$\rho_i^{n+1} = \rho_i^n - \frac{\Delta t v_{}}{\Delta x} \left( \rho_i^n - \rho_{i-1}^n \right) \enspace .$$ (108)

1. Discrete differences:
   • Characteristic curves: mass flows from left to right
   • Replacement: CS of naive scheme $\rightarrow$ difference in upwind direction: BS
2. Characteristics + interpolation:
   • Follow characteristic curve from $\rho_i^{n+1}$ back in time
   • Interpolate between $\rho_{i-1}^n$ and $\rho_i^n$ (linearly)
3. Reconstruct-Solve-Average (RSA, Godunov-type method):
   • Finite volumes (reconstruction-evolution, Reconstruct-Propagate-Average)
   • Reconstruct run of $\rho$ within each cell (here: assume constant $\rho$)
   • Use exact continuous solution for the time evolution
   • Average intermediate continuous result to get single value in each cell