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Professor Sergei K. Godunov's most influential work is in the area of applied and numerical mathematics. It has had a major impact on science and engineering, particularly in the development of methodologies 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.
Godunov's theorem, also known as Godunov's order barrier theorem states that:
The theorem was originally proved by Godunov as a Ph.D. student at Moscow State University and has been extremely important in the development of the theory of high resolution schemes for the numerical solution of PDEs.
Additional recommended knowledge
We generally follow Wesseling (2001).
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
It is assumed 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,
The above scheme of equation (2) is monotonicity preserving if and only if
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.
For the same monotonically increasing , assume that for some and choose
Linear one-step second-order accurate numerical schemes for the convection equation
cannot be monotonicity preserving unless
where is the signed Courant–Friedrichs–Lewy condition (CFL) number.
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.
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Godunov's_theorem". A list of authors is available in Wikipedia.|