In mathematics, the semi-implicit Euler method, also called symplectic Euler, semi-explicit Euler, Euler–Cromer, and Newton–Størmer–Verlet (NSV), is a modification of the Euler method for solving Hamilton's equations, a system of ordinary differential equations that arises in classical mechanics. It is a symplectic integrator and hence it yields better results than the standard Euler method.
Origin The method has been discovered and forgotten many times, dating back to Newton's Principiae, as recalled by Richard Feynman in his Feynman Lectures (Vol. 1, Sec. 9.6) In modern times, the method was rediscovered in a 1956 preprint by René De Vogelaere that, although never formally published, influenced subsequent work on higher-order symplectic methods.
Setting The semi-implicit Euler method can be applied to a pair of differential equations of the form
d x d t = f ( t , v ) d v d t = g ( t , x ) , {\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=f(t,v)\\{\frac {dv}{dt}}&=g(t,x),\end{aligned}}}
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
H = T ( t , v ) + V ( t , x ) . {\displaystyle H=T(t,v)+V(t,x).\,}
The differential equations are to be solved with the initial condition
x ( t 0 ) = x 0 , v ( t 0 ) = v 0 . {\displaystyle x(t_{0})=x_{0},\qquad v(t_{0})=v_{0}.}
The method The semi-implicit Euler method produces an approximate discrete solution by iterating
v n + 1 = v n + g ( t n , x n ) Δ t x n + 1 = x n + f ( t n , v n + 1 ) Δ t {\displaystyle {\begin{aligned}v_{n+1}&=v_{n}+g(t_{n},x_{n})\,\Delta t\\[0.3em]x_{n+1}&=x_{n}+f(t_{n},v_{n+1})\,\Delta t\end{aligned}}}
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