Professor Sergei K. 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.
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 and , such a scheme can be described by
In other words, the solution at time and location is a linear function of the solution at the previous time step . We assume that determines uniquely. Now, since the above equation represents a linear relationship between and 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 .
Then, because it therefore follows that because
This means that monotonicity is preserved for this case.
Case 2: (necessary condition)
We prove the necessary condition by contradiction. Assume that for some 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
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 and choose such that . This implies that and .
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.
Godunov, Sergei K. (1954), Ph.D. Dissertation: Different Methods for Shock Waves, Moscow State University.
Godunov, Sergei K. (1959), A Difference Scheme for Numerical Solution of Discontinuous Solution of Hydrodynamic Equations, Math. Sbornik, 47, 271-306, translated US Joint Publ. Res. Service, JPRS 7226, 1969.
Wesseling, Pieter (2001), Principles of Computational Fluid Dynamics, Springer-Verlag.