Semi-implicit Euler method

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.

Setting

The semi–implicit Euler method can be applied to a pair of differential equations of the form

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

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

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

$H=T(t,v)+V(t,x).\,$

The differential equations are to be solved with the initial condition

$x(t_{0})=x_{0},\qquad v(t_{0})=v_{0}.$

The method

The semi–implicit Euler method produces an approximate discrete solution by iterating

$v_{n+1}=v_{n}+g(t_{n},x_{n})\,\Delta t\quad$

$x_{n+1}=x_{n}+f(t_{n},v_{n+1})\,\Delta t\quad$

where $\Delta t$ is the time step and $t_{n}=t_{0}+n\Delta t$ is the time after n steps.

The difference with the standard Euler method is that the semi–implicit Euler method uses $v_{n+1}$ in the equation for $x_{n+1}$, while the Euler method uses $v_{n}$.

The semi–implicit Euler is a first-order integrator, just as the standard Euler method. This means that it commits a global error of the order of Δt. However, the semi–implicit Euler method is a symplectic integrator, unlike the standard method. As a consequence, the semi–implicit Euler method almost conserves the energy (when the Hamiltonian is time-independent). Often, the energy increases steadily when the standard Euler method is applied, making it far less accurate.

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 semi–implicit Euler for this equation is

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

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

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.
• Vesely, Franz J. (2001). Computational Physics: An Introduction (2nd edition ed.). Springer. pp. page 117. ISBN 978-0-306-46631-1. {{cite book}}: |edition= has extra text (help); |pages= has extra text (help)