Semi-implicit Euler method: Difference between revisions

Revision as of 15:38, 3 March 2007

Revision as of 10:58, 4 March 2007

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

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

:<math> {dv \over dt} = a(x,v,t), </math>

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

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

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

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.

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>.

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== Example ==

The motion of a [[spring (device)|spring]] satisfying [[Hooke's law]] is given by

For a spring undergoing [[simple harmonic motion]] we have the following

:<math>F ={dx -kx\over dt} = mav(t) \quad</math>

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

The Euler-Cromer algorithm for this equation is

Thus we have solutions

:<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 ===

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== References ==

* {{cite book |last= Giordano

|first= Nicholas J.

|coauthors= Hisao Nakanishi

|title= Computational Physics

|edition= 2nd edition

|publisher= Benjamin Cummings

|year= 2005

|month= July

|language= English

|isbn= 0-1314-6990-8 }}

* {{cite web

| firstlast = MacDonald

{{cite web

| lastfirst = James

| first = MacDonald

| authorlink = http://www.physics.udel.edu/~jim

| title = The Euler-Cromer method

| publisher = [[University of Delaware]]