Energy error

For a computed state $z_n$ and an exact state $z(t_n)$, an energy error is $H(z_n)-H(z(t_n))$ for a specified energy function $H$. If the exact dynamics conserve $H$, the reference is simply $H(z(0))$. This is a scalar diagnostic; it does not by itself bound position or phase errors.

For the unit-frequency harmonic oscillator $q^{\prime }=p$, $p^{\prime }=-q$, the exact energy is $H=(q^2+p^2)/2$. The Forward Euler method givesThus a fixed positive step produces systematic energy growth even though the exact oscillator has constant energy. A bounded oscillatory energy error has a different long-time meaning from such cumulative drift. Comparing methods also requires the same step size, reference trajectory and observation interval.