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

${\displaystyle {\begin{aligned}v_{n+1}&=v_{n}+g(t_{n},x_{n})\,\Delta t\\[0.3em]x_{n+1}&=x_{n}+f(t_{n},v_{n+1})\,\Delta t\end{aligned}}}$ 

where 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.

Applying the method with negative time step to the computation of ${\displaystyle (x_{n},v_{n})}$  from ${\displaystyle (x_{n+1},v_{n+1})}$  and rearranging leads to the second variant of the semi-implicit Euler method

${\displaystyle {\begin{aligned}x_{n+1}&=x_{n}+f(t_{n},v_{n})\,\Delta t\\[0.3em]v_{n+1}&=v_{n}+g(t_{n},x_{n+1})\,\Delta t\end{aligned}}}$ 

which has similar properties.

The semi-implicit Euler is a first-order integrator, just as the standard Euler method. 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.

Alternating between the two variants of the semi-implicit Euler method leads in one simplification to the Störmer-Verlet integration and in a slightly different simplification to the leapfrog integration, increasing both the order of the error and the order of preservation of energy.^[1]

Example

The motion of a spring satisfying Hooke's law is given by

${\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=v(t)\\[0.2em]{\frac {dv}{dt}}&=-{\frac {k}{m}}\,x=-\omega ^{2}\,x.\end{aligned}}}$ 

The semi-implicit Euler for this equation is

${\displaystyle {\begin{aligned}v_{n+1}&=v_{n}-\omega ^{2}\,x_{n}\,\Delta t\\[0.2em]x_{n+1}&=x_{n}+v_{n+1}\,\Delta t.\end{aligned}}}$ 

The iteration preserves the modified energy functional ${\displaystyle E_{h}(x,v)={\tfrac {1}{2}}\left(v^{2}+\omega ^{2}\,x^{2}-\omega ^{2}\Delta t\,vx\right)}$  exactly, leading to stable periodic orbits that deviate by ${\displaystyle O(\Delta t)}$  from the exact orbits. The exact circular frequency ${\displaystyle \omega }$  increases in the numerical approximation by a factor of ${\displaystyle 1+{\tfrac {1}{24}}\omega ^{2}\Delta t^{2}+O(\Delta t^{4})}$ .

References

1. Hairer, Ernst; Lubich, Christian; Wanner, Gerhard (2003). "Geometric numerical integration illustrated by the Störmer/Verlet method". Acta Numerica. 12: 399–450. doi:10.1017/S0962492902000144.

• 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)