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

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

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

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

${\displaystyle H=T(t,v)+V(t,x).\,}$ 

The differential equations are to be solved with the initial condition

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

The method

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

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

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

where Δt is the time step and t_n = t₀ + nΔt 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

${\displaystyle {dx \over dt}=v(t)}$ 

${\displaystyle {dv \over dt}=-{k \over m}x.\quad }$ 

The semi-implicit Euler for this equation is

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

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

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)