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.
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.
, such a scheme can be described by In other words, the solution
is a linear function of the solution at the previous time step
uniquely.
Now, since the above equation represents a linear relationship between
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
is monotonically increasing with
because This means that monotonicity is preserved for this case.
Case 2: (necessary condition) We prove the necessary condition by contradiction.
and choose the following monotonically increasing
, Then from equation (2) we get Now choose
, to give which implies that
is NOT increasing, and we have a contradiction.
Thus, monotonicity is NOT preserved for
, 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
σ
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,
Now, it is clear from equation (15) that Assume
σ > 0 ,
It therefore follows that, which contradicts equation (16) and completes the proof.
The exceptional situation whereby
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.