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I am preparing a lecture course on the applications of operator theory where I intended to make some numerical analysis application. I was wondering about this question while browsing the literature I can access.

Lax and Richtmyer (1955) start with a vague reference to Courant, Friedrichs, and Lewy (1928).

Trotter (1958) on the other hand refers to von Neumann, without any specific reference.

Since I have no deeper background in numerical analysis, I might be missing some obvious point. So can someone tell me (with a reference) who introduced the notion of stability in numerical analysis?

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The original German CFL is available at resolver.sub.uni-goettingen.de/purl?GDZPPN002272636 which I found off en.wikipedia.org/wiki/… but as the wikipedia article does not use the word stability anywhere one must wonder. Is there only one notion of stability in numerical analysis, or, more to the point, is there one obvious notion of stability that comes from the operator theory direction? –  Will Jagy Sep 2 '11 at 22:07
    
Will, thanks. It is great to have this paper, and I should have found it myself... A note: Du Fort and Frankel in this paper: ams.org/journals/mcom/1953-07-043/S0025-5718-1953-0059077-7 introduces the notion of stability/instability referring to the CFL condition. This seems to show me that maybe it was not operator theory what motivated this notion. –  András Bátkai Sep 2 '11 at 22:26
    
Wait... are you considering the stability of a numerical algorithm, or some other more specialized concept of stability? –  J. M. Sep 3 '11 at 6:38
    
Well, maybe I was not clear enough because I do not know the subject well. Specifically, my motivation was the concept in the Lax equivalence theorem. Clearly, someone must have introduced the notion of stability for a finite difference scheme well before, probably based on CFL. Who and where? –  András Bátkai Sep 3 '11 at 7:41

1 Answer 1

up vote 13 down vote accepted

John von Neumann is credited as having pioneered the stability analysis of finite difference schemes. Crank and Nicholson [1] acknowledge Von Neumann when they demonstrate the stability of their scheme in 1947, and a few years later the method was applied in a meteorological context in a paper co-authored by Von Neumann [2]. That 1950 paper introduces the stability analysis as being "patterned after the rigorous method of Courant, Friedrichs, and Lewy."

[1] J. Crank and P. Nicolson, Proc. Camb. Phil. Soc. 43, 50–67 (1947).

[2] J.G. Charney, R. Fjörtoft, and J. von Neumann, Tellus 2, 237-254 (1950). Online at http://mathsci.ucd.ie/~plynch/eniac/CFvN-1950.pdf

Further reading:

http://en.wikipedia.org/wiki/Von_Neumann_stability_analysis

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Thank you. This is the answer I was looking for. –  András Bátkai Sep 4 '11 at 23:05

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