Godunov's theorem

Remarks

The conditions for $m = 0, 1, 2$ say that the scheme is exact on polynomials of degree at most $2$, which is the same as a local truncation error $O(\Delta x^3)$ for smooth solutions. Proof: the numbers $c_k \ge 0$ sum to $1$, so they are the weights of a probability distribution on the integers $k$, with mean $-\nu$ and second moment $\nu^2$. So its variance is $0$, and all the weight sits on the single point $k = -\nu$, which must then be an integer. For a linear scheme, monotone is the same as monotonicity preserving: an increasing step $u_i = 1$ for $i > J$ and $0$ otherwise gives $u_{j+1}^{n+1} - u_j^{n+1} = c_{J-j}$. Examples: the upwind scheme with $0 \le \nu \le 1$ is monotone and of first order. The Lax–Wendroff method is of second order, has the negative coefficient $c_1 = -\nu(1 - \nu)/2$ for $0 < \nu < 1$, and oscillates near jumps.

Going deeper

For a student

Godunov proved this in 1959, and it is the reason why modern high-resolution schemes are nonlinear. Total variation diminishing schemes, flux limiters, MUSCL and ENO/WENO schemes switch to a first-order monotone flux near discontinuities and use a higher-order flux in smooth regions. The theorem is a statement about linear schemes; it does not limit nonlinear ones, which can be second order away from extrema.