Semi-implicit Euler method: Difference between revisions

Revision as of 05:17, 23 October 2022

Revision as of 21:31, 3 December 2022

Line 3:

== 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>{{citation needed|date=September 2019}}\\

{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 31:

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>

Line 69:

\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>.

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 ==

Line 90 ⟶ 89:

}}</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 }}