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Home , Hans Lewy, Kurt Otto Friedrichs, Richard Courant

“On the Partial Difference Equations of ” R Courant, K Friedrichs, H Lewy

TRANSCENDING DISCIPLINES, TRANSFORMING LIVES Authors

2 • 1888-1972, Polish • Born in Lulinitz (modern-day ) • Earned PhD under at University of Göttingen • Numerous contributions to • NYU Courant Institute named after him: cofounded with Friedrichs • “Empirical evidence can never establish mathematical existence--nor can the 's demand for existence be dismissed by the physicist as useless rigor. Only a mathematical existence proof can ensure that the mathematical description of a physical phenomenon is meaningful.” - Courant, quoted in The Parsimonious Universe. • NYTimes Obituary: https://www.nytimes.com/1972/01/29/archives/ dr-richard-courant-dies-at-84-influential--scholar.html

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• 1901-1982, German

• Born in Schleswig-Holstein (Denmark/)

• Grew up in Düsseldorf, Germany.

• Earned doctorate 1927 under Courant at Göttingen

• Numerous contributions to mathematical analysis

• Doctoral students include , and…

Fu Foundation Professor Emeritus of Chia-Kun Chu!

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• 1904-1988 German-born American

• Born in Breslau, Germany (Wrocław, Poland)

• Earned doctorate 1926 under Courant at Göttingen Photo by George Bergman

• Studied under Max Brown, Courant, , David Hilbert, , , and ..

• Göttingen was a hotbed for mathematical analysis, clearly!

5 The Paper

6 Roots and Historical Significance

• Paper grew out of research Friedrichs had done with Lewy (1927) on initial value problems from linear hyperbolic partial differential equations. • Discovered that step sizes in time could not be taken arbitrary but must be smaller than a constant times the step size of the space variable. • The derived bound is known as the CFL condition.

• “…There is scarcely a talk or a paper on modeling phenomena governed by so-called explicit difference schemes where this number [CFL] does not come up.

Friedrichs was interested, however, in proving existence theorems for partial differential equations by letting the mesh size and time step become vanishing small. The modeling aspect was a side product that only became important after World War II when one could compute something useful on a large computer. In later life, when Friedrichs was pressed to say something about the important role of computer modeling in applied mathematics to which he had made such a fundamental contribution, he simply would not bite.” —National Academy of Sciences. 1995. Biographical Memoirs: V.67. Washington, DC: The National Academies Press. https://doi.org/10.17226/4894. https://www.nap.edu/read/4894/chapter/8#133

7 On to Paper!

[The[The speaker speaker wants wants to to acknowledge acknowledge support support generously generously provided provided by by the the NSF NSF GRFP GRFP under under Grant Grant #DGE #DGE - 1644869!]- 1644869!] 8 Sources

• Hildebrandt, S and Tromba, A. The Parsimonious Universe: Shape and Form in the Natural World. Springer Science & Business Media, 1996.

• National Academy of Sciences. 1995. Biographical Memoirs: V.67. Washington, DC: The National Academies Press. https://doi.org/10.17226/4894. https://www.nap.edu/read/ 4894/chapter/8#133

• This reference is noted in the historical roots and significance of the paper.

• Wikipedia contributors. "Richard Courant." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 2 Oct. 2018. Web. 5 Oct. 2018. https://en.wikipedia.org/wiki/Richard_Courant

• Wikipedia contributors. "Kurt Otto Friedrichs." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 23 Jun. 2018. Web. 5 Oct. 2018. https://en.wikipedia.org/wiki/Kurt_Otto_Friedrichs

• Wikipedia contributors. "Hans Lewy." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 16 May. 2018. Web. 5 Oct. 2018. https://en.wikipedia.org/wiki/Hans_Lewy

• Photo of Courant is by Konrad Jacobs, Erlangen and copyrighted under the Mathematisches Forschungsinstitut Oberwolfach license, which may be used on terms of the Creative Commons License Attribution-Share Alike 2.0 Germany. (From website https://opc.mfo.de accessed 10/05/2018).

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