Semi-implicit Euler method: Difference between revisions

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In mathematics, the '''Euler-CromerEuler–Cromer algorithm''' or '''symplectic Euler method''' is a modification of the [[Euler integration|Euler method]] for solving [[Hamilton's equations]], a system of [[ordinary differential equation]]s. Itthat givesarises muchin better[[classical resultsmechanics]]. forIt is a [[Oscillation|oscillatorysymplectic integrator]] solutionsand hence it yields better results than the standard Euler method.

== Setting ==

Given a pair of [[differential equation]]s of the form

GivenThe Euler–Cromer algorithm can be applied to a pair of [[differential equation]]s of the form

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

:<math> {dvdx \over dt} = af(xt,v,t), </math>

:<math> {dxdv \over dt} = vg(t,x), </math>

where ''a'' is a given function, and initial conditions

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

:<math> (x_0,v_0), \quad</math>

:<math> H = T(x_0t,v_0v) + V(t,x). \quad, </math>

the Euler-Cromer algorithm produces an approximate [[Discrete mathematics|discrete]] solution by iterating

The differential equations are to be solved with the initial condition

:<math> v_{n+1} = v_n + a_n \Delta t \quad</math>

:<math> x_{n+1}x(t_0) = x_nx_0, +\qquad v_{n+1}v(t_0) \Delta= tv_0. \quad</math>

== SeeThe alsomethod ===

where <math> \Delta t </math> is the timestep and <math>a_n = a(x_n,v_n,t_n)</math> is the acceleration at the current timestep.

theThe Euler-CromerEuler–Cromer algorithm produces an approximate [[Discrete mathematics|discrete]] solution by iterating

Note the difference from the Euler method: <math>x_{n+1}</math> depends on <math>v_{n+1}</math> rather than <math>v_n</math>.

:<math> v_{n+1} = v_n + a_nf(t_n, x_n) \, \Delta t \quad</math>

:<math> x_{n+1} = x_n + g(t_n, v_{n+1}) \, \Delta t \quad</math>

where <math> \Delta t </math> is the timesteptime step and <math>a_nt_n = a(x_n,v_n,t_n)t_0 + n\Delta t</math> is the accelerationtime at theafter current''n'' timestepsteps.

Note theThe difference fromwith the standard Euler method: is that the Euler–Cromer method uses <math>x_v_{n+1}</math> dependsin onthe equation for <math>v_x_{n+1}</math>, ratherwhile the Euler method thanuses <math>v_n</math>.

The Euler–Cromer method is a [[Numerical ordinary differential equations#Consistency and order|first-order integrator]], just as the standard Euler method. This means that it commits an global error of the order of &Delta;t. However, the Euler–Cromer method is a [[symplectic integrator]], unlike the standard method. As a consequence, the Euler–Cromer method almost conserves the energy (when the Hamiltonian is time-independent). Often, the energy decreases steadily when the standard Euler method is applied, making it far less accurate.

== Example ==

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:<math> {dv \over dt} = -{k \over m}x. \quad</math>

The Euler-CromerEuler–Cromer algorithm for this equation is

:<math>v_{n+1} = v_n - {k \over m}x_n\Delta t \quad</math>

:<math>x_{n+1} = x_n + v_{n+1} \Delta t. \quad</math>

== See also ===

* [[Numerical ordinary differential equations]]

== References ==

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

| accessdate = 2007-03-03}}

* {{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}}

{{mathapplied-stub}}

[[Category:Numerical differential equations]]