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

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

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

and initial conditions

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

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

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

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

where ${\displaystyle \Delta t}$ is the timestep.

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

See also =

• Numerical ordinary differential equations

References

Giordano, Nicholas J. (2005). Computational Physics (2nd ed.). Benjamin Cummings. ISBN 0-1314-6990-8. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)

James, MacDonald. "The Euler-Cromer method". University of Delaware. Retrieved 2007-03-03. {{cite web}}: Check |authorlink= value (help); External link in |authorlink= (help)