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.

See also =

• Numerical ordinary differential equations