  This paper examines a class of PDEs where some part of the PDE system evolves
a vector field whose curl remains zero or grows in proportion to specified
source terms. Such PDEs are referred to as curl-free or curl-preserving
respectively. In this paper we catalogue a class of DG-like schemes for such
PDEs. To retain the globally curl-free or curl-preserving constraints, the
components of the vector field, as well as their higher moments, have to be
collocated at the edges of the mesh. They are updated by using potentials that
are collocated at the vertices of the mesh. The resulting schemes : 1) do not
blow up even after very long integration times, 2) do not need any special
cleaning treatment, 3) can operate with large explicit timesteps, 4) do not
require the solution of an elliptic system and 5) can be extended to higher
orders using DG-like methods. The methods rely on a special curl-preserving
reconstruction and they also rely on multidimensional upwinding. The Galerkin
projection, so crucial to the design of a DG method, is now carried out in the
edges of the mesh and yields a weak form update that uses potentials that are
obtained at the vertices of the mesh with the help of a multidimensional
Riemann solver. A von Neumann stability analysis of the curl-preserving methods
is carried out and the limiting CFL numbers of this entire family of methods is
catalogued in this work. The stability analysis confirms that with increasing
order of accuracy, our novel curl-free methods have superlative phase accuracy
while substantially reducing dissipation. We also show that PNPM-like methods
retain much of the excellent wave propagation characteristics of the DG-like
methods while offering a much larger CFL number and lower computational
complexity.
