Godunov's theorem

The theorem states that:

Linear numerical schemes for solving partial differential equations (PDE's), having the property of not generating new extrema (monotone scheme), can be at most first-order accurate.

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 ${\displaystyle x_j=j\mathrm{\Delta }x}$ and ${\displaystyle t^n=n\mathrm{\Delta }t}$, such a scheme can be described by

In other words, the solution ${\displaystyle {\phi }_j^{n+1}}$ at time ${\displaystyle n+1}$ and location ${\displaystyle j}$ is a linear function of the solution at the previous time step ${\displaystyle n}$. We assume that ${\displaystyle {\beta }_m}$ determines ${\displaystyle {\phi }_j^{n+1}}$ uniquely. Now, since the above equation represents a linear relationship between ${\displaystyle {\phi }_j^n}$ and ${\displaystyle {\phi }_j^{n+1}}$ 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 ${\displaystyle {\phi }_j^n}$ is monotonically increasing with ${\displaystyle j}$.

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

This means that monotonicity is preserved for this case.

Case 2: (necessary condition)

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

Then from equation (2) we get

Now choose ${\displaystyle j=k-p}$, to give

which implies that ${\displaystyle {\phi }_j^{n+1}}$ is NOT increasing, and we have a contradiction. Thus, monotonicity is NOT preserved for ${\displaystyle {\gamma }_p<0}$, 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 ${\displaystyle \sigma }$ 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, ${\displaystyle {\gamma }_m\ge 0}$.

Now, it is clear from equation (15) that

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

It therefore follows that,

which contradicts equation (16) and completes the proof.

The exceptional situation whereby ${\displaystyle \sigma =\left|c\right|\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}\in \mathbb{N}}$ 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.