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 ==

The semi-implicit Euler method can be applied to a pair of [[differential equation]]s of the form{{citation needed|date=September 2019}}

:<math>\begin{align}

:<math> {dx \over dt} &= f(t,v) </math>\\

{dv \over dt} &= g(t,x),

:<math> {dv \over dtend{align} = g(t,x), </math>

where ''f'' and ''g'' are given functions. Here, ''x'' and ''v'' may be either scalars or vectors. The equations of motion in [[Hamiltonian mechanics]] take this form if the Hamiltonian is of the form

Line 25 ⟶ 29:

\end{align}</math>

where Δ''t'' is the time step and ''t_n'' = ''t''₀'' + ''n''Δ''t'' is the time after ''n'' steps.

The difference with the standard Euler method is that the semi-implicit Euler method uses ''v''_''n''+1 in the equation for ''x''_''n''+1, while the Euler method uses ''v_n''.

Applying the method with negative time step to the computation of <math>(x_n, v_n)</math> from <math>(x_{n+1}, v_{n+1})</math> and rearranging leads to the second variant of the semi-implicit Euler method

:<math>\begin{align}

x_{n+1} &= x_n + f(t_n, v_n) \, \Delta t\\[0.3em3ex]

v_{n+1} &= v_n + g(t_n, x_{n+1}) \, \Delta t

\end{align}</math>

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Alternating between the two variants of the semi-implicit Euler method leads in one simplification to the Störmer-[[Verlet integration]] and in a slightly different simplification to the [[leapfrog integration]], increasing both the order of the error and the order of preservation of energy.<ref name="hairer2003" />

The stability region of the semi-implicit method was presented inby Niiranen<ref>[httphttps://catwww.inistresearchgate.frnet/publication/?aModele=afficheN&cpsidt=1008675268034494_Fast_and_accurate_symmetric_Euler_algorithm_for_electromechanical_simulations_NOTE_The_method_became_later_known_as_Symplectic_Euler Niiranen, J.Jouko: Fast and accurate symmetric Euler algorithm for electromechanical simulations] Proceedings of the Electrimacs'99, Sept. 14-16, 1999 Lisboa, Portugal, Vol. 1, pages 71 - 78.</ref> although the methodsemi-implicit Euler was misleadingly called symmetric Euler in his paper. The semi-implicit method models the simulated system correctly if the complex roots of the characteristic equation are within the circle shown below. For real roots the stability region extends outside the circle for which the criteriacriterion is <math>s > - 2/\Delta t</math>

[[Image:Symplectic Euler stability region.jpeg]]

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\end{align}</math>

Substituting <math>v_{n+1}</math> in the second equation with the expression given by the first equation, the iteration can be expressed in the following matrix form

The iteration preserves the modified energy functional <math>E_h(x,v)=\tfrac12\left(v^2+\omega^2\,x^2-\omega^2\Delta t\,vx\right)</math> exactly, leading to stable periodic orbits that deviate by <math>O(\Delta t)</math> from the exact orbits. The exact circular frequency <math>\omega</math> increases in the numerical approximation by a factor of <math>1+\tfrac1{24}\omega^2\Delta t^2+O(\Delta t^4)</math>.

:<math>\begin{bmatrix} x_{n+1} \\v_{n+1}\end{bmatrix} =

\begin{bmatrix}

1-\omega^2 \Delta t^2 & \Delta t \\

-\omega^2 \Delta t & 1

\end{bmatrix} \begin{bmatrix} x_{n} \\ v_{n} \end{bmatrix},</math>

and since the [[determinant]] of the matrix is 1 the transformation is area-preserving.

The iteration preserves the modified energy functional <math>E_h(x,v)=\tfrac12\left(v^2+\omega^2\,x^2-\omega^2\Delta t\,vx\right)</math> exactly, leading to stable periodic orbits (for sufficiently small step size) that deviate by <math>O(\Delta t)</math> from the exact orbits. The exact circular frequency <math>\omega</math> increases in the numerical approximation by a factor of <math>1+\tfrac1{24}\omega^2\Delta t^2+O(\Delta t^4)</math>.

== References ==

<references>

<ref name="hairer2003">{{cite journal

| firstfirst1=Ernst | lastlast1=Hairer

| first2=Christian | last2=Lubich

| first3=Gerhard | last3=Wanner

Line 75 ⟶ 87:

| volume = 12

| pages = 399–450

| url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.7.7106

| doi=10.1017/S0962492902000144

| bibcode=2003AcNum..12..399H

| citeseerx=10.1.1.7.7106

| 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 = MacDonaldFeynman

| first = JamesRichard 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 book |last= Giordano

* {{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}}

|first= Nicholas J.

* {{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]}}

|coauthors= Hisao Nakanishi

* {{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 }}

|title= Computational Physics

|edition= 2nd edition

|publisher= Benjamin Cummings

|date=July 2005

|isbn= 0-13-146990-8 }}

* {{cite web

| last = MacDonald

| first = James

| title = The Euler-Cromer method

| publisher = [[University of Delaware]]

| url = http://www.physics.udel.edu/~jim/PHYS460_660_13S/Ordinary%20Differential%20Equations/Euler-Cromer%20Method.htm

| accessdate = 2013-04-11}}

* {{cite book |last= Vesely

|first= Franz J.

|title= Computational Physics: An Introduction

|edition= 2nd edition

|publisher= Springer

|year= 2001

|isbn= 978-0-306-46631-1

|pages=page 117}}

{{Numerical integrators}}