Godunov's theorem

In numerical analysis and computational fluid dynamics, Godunov's theorem — also known as Godunov's order barrier theorem — is a mathematical theorem important in the development of the theory of high-resolution schemes for the numerical solution of partial differential equations.

The theorem states that:

Professor Sergei Godunov originally proved the theorem as a Ph.D. student at Moscow State University. It is his most influential work in the area of applied and numerical mathematics and has had a major impact on science and engineering, particularly in the development of methods used in computational fluid dynamics (CFD) and other computational fields. One of his major contributions was to prove the theorem (Godunov, 1954; Godunov, 1959), that bears his name.

The theorem
We generally follow Wesseling (2001).

Aside

Assume a continuum problem described by a PDE is to be computed using a numerical scheme based upon a uniform computational grid and a one-step, constant step-size, M grid point, integration algorithm, either implicit or explicit. Then if $$ x_{j} = j\,\Delta x $$ and $$t^n  = n\,\Delta t $$, such a scheme can be described by

In other words, the solution $$\varphi _j^{n + 1} $$ at time $$n + 1$$ and location $$j$$ is a linear function of the solution at the previous time step $$n$$. We assume that $$\beta _m $$ determines $$\varphi _j^{n + 1} $$ uniquely. Now, since the above equation represents a linear relationship between $$ \varphi _j^{n } $$ and $$ \varphi _j^{n + 1} $$ we can perform a linear transformation to obtain the following equivalent form,

Theorem 1: Monotonicity preserving

The above scheme of equation (2) is monotonicity preserving if and only if

Proof - Godunov (1959)

Case 1: (sufficient condition)

Assume (3) applies and that $$\varphi _j^n $$ is monotonically increasing with $$j $$.

Then, because $$\varphi _j^n \le \varphi _{j + 1}^n  \le \cdots  \le \varphi _{j + m}^n $$ it therefore follows that $$\varphi _j^{n + 1}  \le \varphi _{j + 1}^{n + 1} \le \cdots  \le \varphi _{j + m}^{n + 1} $$ because

This means that monotonicity is preserved for this case.

Case 2: (necessary condition)

We prove the necessary condition by contradiction. Assume that $$\gamma _p^{} < 0 $$ for some $$p $$ and choose the following monotonically increasing $$\varphi_j^n \, $$,

Then from equation (2) we get

Now choose $$ j = k - p $$, to give

which implies that $$\varphi _j^{n + 1} $$ is NOT increasing, and we have a contradiction. Thus, monotonicity is NOT preserved for $$\gamma _p < 0 $$, which completes the proof.

Theorem 2: Godunov’s Order Barrier Theorem

Linear one-step second-order accurate numerical schemes for the convection equation

cannot be monotonicity preserving unless

where $$ \sigma $$ is the signed Courant–Friedrichs–Lewy condition (CFL) number.

Proof - Godunov (1959)

Assume a numerical scheme of the form described by equation (2) and choose

The exact solution is

If we assume the scheme to be at least second-order accurate, it should produce the following solution exactly

Substituting into equation (2) gives:

Suppose that the scheme IS monotonicity preserving, then according to the theorem 1 above, $$\gamma _m \ge 0 $$.

Now, it is clear from equation (15) that

Assume $$\sigma > 0, \quad \sigma  \notin \mathbb{ N} $$ and choose $$j $$ such that $$ j > \sigma  > \left( j - 1 \right) $$. This implies that $$\left( {j - \sigma } \right) > 0 $$ and $$\left( {j - \sigma - 1} \right) < 0 $$.

It therefore follows that,

which contradicts equation (16) and completes the proof.

The exceptional situation whereby $$\sigma = \left| c \right|{{\Delta t} \over {\Delta x}} \in \mathbb{N} $$ is only of theoretical interest, since this cannot be realised with variable coefficients. Also, integer CFL numbers greater than unity would not be feasible for practical problems.