Numerical volume preservation of a divergence free fluid under symmetry

This paper studies conservative formulations for the evolution of flows in 3-D which satisfy a symmetry condition. It is shown how the Hamiltonian form leads to a proper form of the evolution equation. This is illustrated by a number of examples. It is also shown how this results in conservation of e.g. mass when using simplectic integrators. A special section is devoted to computing displacements by the implicit midpoint rule when the velocities are not available in explicit form but e.g. found from a numerical method like FEM.