Semi-implicit Euler Method - Example

Example

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

\begin{align} \frac{dx}{dt} &= v(t)\\ \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 \\ x_{n+1} &= x_n + v_{n+1} \,\Delta t. \end{align}

The iteration preserves the modified energy functional exactly, leading to stable periodic orbits that deviate by from the exact orbits. The exact circular frequency increases in the numerical approximation by a factor of .

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