In fluid dynamics, it is common to encounter velocity fields that can be written in terms of a streamfunction. In two dimensions,
$$ \dot x = \dfrac{\partial{\psi(x, y)}}{\partial{y}}, \;\;\;\dot y = -\dfrac{\partial{\psi(x, y)}}{\partial{x}} $$
Typically, one can find trajectories subject to $\psi$ via numerical integration of the above velocity field. However, if one uses standard stepping methods, the resulting trajectories will depart from true streamlines, which should correspond to isosurfaces of $\psi$
This appears to me to be a problem suited to a symplectic integrator, since the streamfunction serves a similar role to a Hamiltonian. However, all symplectic integrators of which I am aware take advantage of the fact that one of the Hamiltonian coordinates is the momentum for the other coordinate, which is not true for a streamfunction problem.
Is there a particular type of numerical integration scheme that can conserve $\psi$ for this sort of problem? As an extension, if the streamfunction is explicitly time dependent ($\psi(t)$), can the same method be used to ensure that $\psi(t)$ is conserved instantaneously? What if we restrict $\psi(t)$ to be periodic?