Semi-implicit Euler method: Difference between revisions

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{{Short description|Modification of the Euler method for solving Hamilton's equations}}
In mathematics, the '''semi-implicit Euler method''', also called '''symplectic Euler''', '''semi-explicit Euler''', '''Euler–Cromer''', and '''Newton–Størmer–Verlet (NSV)''', is a modification of the [[Euler integration|Euler method]] for solving [[Hamilton's equations]], a system of [[ordinary differential equation]]s that arises in [[classical mechanics]]. It is a [[symplectic integrator]] and hence it yields better results than the standard Euler method.

== 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" />

== 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| title = On the famous unpublished preprint "Methods of integration which preserve the contact transformation property of the Hamilton equations" by René De Vogelaere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|url-status=live|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]}}

* {{cite book |last= Giordano|first= Nicholas J.|author2=Hisao Nakanishi|title= Computational Physics|edition= 2nd|publisher= Benjamin Cummings|date=July 2005|isbn= 0-13-146990-8 }}