Energy drift

Remarks

Example: for $H(q, p) = \frac12 (q^2 + p^2)$, the explicit Euler method $q_{n+1} = q_n + h p_n$, $p_{n+1} = p_n - h q_n$ gives $q_{n+1}^2 + p_{n+1}^2 = (1 + h^2)(q_n^2 + p_n^2)$. So $H(y_n) = (1 + h^2)^n H(y_0)$, and the energy error grows exponentially in $n$: the method has energy drift. Non-example: the symplectic Euler method $p_{n+1} = p_n - h q_n$, $q_{n+1} = q_n + h p_{n+1}$ is a linear map of determinant $1$ and trace $2 - h^2$. For $0 < h < 2$ its eigenvalues are distinct and of modulus $1$, so the sequence $y_n$ is bounded and so is $e_n$. There is no drift, only a bounded oscillation of size $O(h)$. More generally, let $H$ be analytic and use a symplectic integrator of order $p$. If the numerical solution stays in a compact set, backward error analysis gives $e_n = O(h^p)$ over times exponentially long in $1/h$. In floating-point arithmetic, round-off adds a slow random-walk drift even to such methods.