Semi-implicit Euler method

EN Texto en inglés Aún no hay artículo en tu idioma: extracto en inglés.

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. 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. 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 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}}} where Δ t {\displaystyle \Delta t} is the time step and t n = t 0 + n Δ t {\displaystyle t_{n}=t_{0}+n\Delta t} is the time after n {\displaystyle n} steps. The difference with the standard Euler method is that the semi-implicit Euler method uses v n + 1 {\displaystyle v_{n+1}} in the equation for x n + 1 {\displaystyle x_{n+1}} , while the Euler method uses v n {\displaystyle v_{n}} . Applying the method with negative time step to the computation of ( x n , v n ) {\displaystyle (x_{n},v_{n})} from ( x n + 1 , v n + 1 ) {\displaystyle (x_{n+1},v_{n+1})} and rearranging leads to the second variant of the semi-implicit Euler method x n + 1 = x n + f ( t n , v n ) Δ t v n + 1 = v n + g ( t n , x n + 1 ) Δ t {\displaystyle {\begin{aligned}x_{n+1}&=x_{n}+f(t_{n},v_{n})\,\Delta t\\[0.3ex]v_{n+1}&=v_{n}+g(t_{n},x_{n+1})\,\Delta t\end{aligned}}} which has similar properties. The semi-implicit Euler is a first-order integrator, just as the standard Euler method.