Semi-implicit Euler method: Difference between revisions

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== Origin ==

The method has been discovered and forgotten many times, dating back to Newton's ''Principiae'',<ref name="hairer2003" /> as recalled by Richard Feynman in his ''Feynman Lectures'' (Vol. 1, Sec. 9.6)<ref name="feynman1963" /> In modern times, the method was rediscovered in a 1956 preprint by René De Vogelaere that, although never formally published, influenced subsequent work on higher-order symplectic methods.<ref name="skeel2003" />

The method was accidentally discovered by [[Newton North High School]] senior student Abby Aspel in 1980. In a lab assignment simulating orbits using Kepler's Law, which required computation in [[BASIC]]: she accidentally reversed two lines of code by calculating velocity before position. Her simulation converged more quickly and resulted in more accurate and feasible results than what was expected. Alan Cromer then proved why her algorithm was more stable than previous methods of computation.<ref>{{Cite journal |last=Cromer |first=Alan |date=1981-05-01 |title=Stable solutions using the Euler approximation |url=https://doi.org/10.1119/1.12478 |journal=American Journal of Physics |volume=49 |issue=5 |pages=455–459 |doi=10.1119/1.12478 |issn=0002-9505}}</ref>

== Setting ==

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| s2cid=122016794

}}</ref>

<ref name="skeel2003">{{cite arXiv

| last1 = Skeel

| first1 = Robert D.

| last2 = Cieśliński

| first2 = Jan L.

| date = 2020

| title = On the famous unpublished preprint "Methods of integration which preserve the contact transformation property of the Hamilton equations" by René De Vogelaere

| class = math.NA

| eprint = 2003.12268

}}</ref>

<ref name="feynman1963">{{cite book

| last = Feynman

| first = Richard P

| author-link =

| date = 1963

| title = ''The Feynman Lectures on Physics'', Vol. 1, Sec. 9.6

| url = https://www.feynmanlectures.caltech.edu/I_09.html

}}</ref>

</references>

* {{cite web|last=Nikolic|first=Branislav K.|title=Euler-Cromer method|publisher=[[University of Delaware]] | url=http://www.physics.udel.edu/~bnikolic/teaching/phys660/numerical_ode/node2.html|accessdate=2021-09-29}}

* {{cite book |last= Vesely|first= Franz J.|title= Computational Physics: An Introduction|url= https://archive.org/details/computationalphy00vese_143|url-access= limited|edition= 2nd|publisher= Springer|year= 2001|isbn= 978-0-306-46631-1|pages=[https://archive.org/details/computationalphy00vese_143/page/n125 117]}}