Godunov's scheme

Remarks

Let $u^*(u_l, u_r)$ be the value at $x/t = 0$ of the Riemann solution with states $u_l, u_r$. Integrating the equation over $C_j \times (t_n, t_{n+1})$ gives the conservation form $$u_j^{n+1} = u_j^n - \lambda \left( F(u^*(u_j^n, u_{j+1}^n)) - F(u^*(u_{j-1}^n, u_j^n)) \right).$$ In this form the scheme only needs $\Lambda \lambda \le 1$. For a scalar law the numerical flux is $\min_{u_l \le w \le u_r} F(w)$ if $u_l \le u_r$, and $\max_{u_r \le w \le u_l} F(w)$ if $u_l > u_r$.

Example: for $u_t + a u_x = 0$ with $a > 0$, the Riemann solution at $x/t = 0$ is $u_l$. So the flux at $x_{j+1/2}$ is $a u_j^n$, and the scheme is the upwind scheme $u_j^{n+1} = u_j^n - a \lambda (u_j^n - u_{j-1}^n)$.

Non-example: the downwind scheme $u_j^{n+1} = u_j^n - a\lambda(u_{j+1}^n - u_j^n)$ uses the flux $a u_{j+1}^n$ instead. This is not the Riemann solution value, and the scheme is unstable for every $\lambda > 0$ when $a > 0$.