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

$\begin{align} v_{n+1} &= v_n + g(t_n, x_n) \, \Delta t\\[0.3em] x_{n+1} &= x_n + f(t_n, v_{n+1}) \, \Delta t \end{align}$

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 (x_n,v_n) from (x_{n + 1},v_{n + 1}) and rearranging leads to the second variant of the semi-implicit Euler method

$\begin{align} x_{n+1} &= x_n + f(t_n, v_n) \, \Delta t\\[0.3em] v_{n+1} &= v_n + g(t_n, x_{n+1}) \, \Delta t \end{align}$

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

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

The semi-implicit Euler for this equation is

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

The iteration preserves the modified energy functional $E_h(x,v)=\tfrac12\left(v^2+\omega^2\,x^2-\omega^2\Delta t\,vx\right)$ exactly, leading to stable periodic orbits that deviate by O(Δt) from the exact orbits. The exact circular frequency ω increases in the numerical approximation by a factor of $1+\tfrac1{24}\omega^2\Delta t^2+O(\Delta t^4)$.

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. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.7.7106. 

• Giordano, Nicholas J.; Hisao Nakanishi (July 2005). Computational Physics (2nd edition ed.). Benjamin Cummings. ISBN 0-1314-6990-8. 
• MacDonald, James. "The Euler-Cromer method". University of Delaware. http://www.physics.udel.edu/~jim/Ordinary%20Differential%20Equations/Euler-Cromer%20Method.htm. 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.