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This paper examines a class of PDEs where some part of the PDE system evolvesa vector field whose curl remains zero or grows in proportion to specifiedsource terms. Such PDEs are referred to as curl-free or curl-preservingrespectively. In this paper we catalogue a class of DG-like schemes for suchPDEs. To retain the globally curl-free or curl-preserving constraints, thecomponents of the vector field, as well as their higher moments, have to becollocated at the edges of the mesh. They are updated by using potentials thatare collocated at the vertices of the mesh. The resulting schemes : 1) do notblow up even after very long integration times, 2) do not need any specialcleaning treatment, 3) can operate with large explicit timesteps, 4) do notrequire the solution of an elliptic system and 5) can be extended to higherorders using DG-like methods. The methods rely on a special curl-preservingreconstruction and they also rely on multidimensional upwinding. The Galerkinprojection, so crucial to the design of a DG method, is now carried out in theedges of the mesh and yields a weak form update that uses potentials that areobtained at the vertices of the mesh with the help of a multidimensionalRiemann solver. A von Neumann stability analysis of the curl-preserving methodsis carried out and the limiting CFL numbers of this entire family of methods iscatalogued in this work. The stability analysis confirms that with increasingorder of accuracy, our novel curl-free methods have superlative phase accuracywhile substantially reducing dissipation. We also show that PNPM-like methodsretain much of the excellent wave propagation characteristics of the DG-likemethods while offering a much larger CFL number and lower computationalcomplexity.

Von Neumann Stability Analysis of DG-like and PNPM-like Schemes for PDEs that have Globally Curl-Preserving Evolution of Vector Fields