2.6.1 Godunov's idea

Discretization paradigms (see Sect. 2.3.16):

• Finite differences: replace derivatives by differences
• Characteristics: follow characteristics back in time and interpolate at old time-level
• Reconstruct-Solve-Average (RSA): Godunov-type finite-volume scheme

RSA: three steps:

• Reconstruct $\rho(x)$ from $\rho_i$.
• Solve the exact problem for $\Delta t$: shift reconstructed function.
• Average $\rho(x)$ in each cell to get $\rho_i$.

Note: Exact solution (and reconstruction) can handle shocks $\Rightarrow$ numerical scheme can handle shocks.