Semi-implicit Euler method

In mathematics, the Euler-Cromer algorithm is a modification of the Euler method for solving ordinary differential equations. It gives much better results for oscillatory solutions.

Given a pair of differential equations of the form

${dx \over dt}=v(t)$

${dv \over dt}=a(x,v,t),$

where a is a given function, and initial conditions

$(x_{0},v_{0}),\quad$

the Euler-Cromer algorithm produces an approximate discrete solution by iterating

$v_{n+1}=v_{n}+a_{n}\Delta t\quad$

$x_{n+1}=x_{n}+v_{n+1}\Delta t\quad$

where $\Delta t$ is the timestep and $a_{n}=a(x_{n},v_{n},t_{n})$ is the acceleration at the current timestep.

Note the difference from the Euler method: $x_{n+1}$ depends on $v_{n+1}$ rather than $v_{n}$.

Example

The motion of a spring satisfying Hooke's law is given by

${dx \over dt}=v(t)$

${dv \over dt}=-{k \over m}x.\quad$

The Euler-Cromer algorithm for this equation is

$v_{n+1}=v_{n}-{k \over m}x_{n}\Delta t\quad$

$x_{n+1}=x_{n}+v_{n+1}\Delta t.\quad$

See also =

• Numerical ordinary differential equations

References

• Giordano, Nicholas J. (2005). Computational Physics (2nd edition ed.). Benjamin Cummings. ISBN 0-1314-6990-8. {{cite book}}: |edition= has extra text (help); Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)
• MacDonald, James. "The Euler-Cromer method". University of Delaware. Retrieved 2007-03-03.