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Several important PDE systems, like magnetohydrodynamics and computationalelectrodynamics, are known to support involutions where the divergence of avector field evolves in divergence-free or divergence constraint-preservingfashion. Recently, new classes of PDE systems have emerged for hyperelasticity,compressible multiphase flows, so-called first order reductions of the Einsteinfield equations, or a novel first order hyperbolic reformulation ofSchr\"odinger's equation, to name a few, where the involution in the PDEsupports curl-free or curl constraint-preserving evolution of a vector field.Since mimetic numerical schemes for the solution of the former class of PDEsare well-developed, we draw guidance from them for the solution of the latterclass of PDEs. We show that a study of the curl constraint-preservingreconstruction gives us a great deal of insight into the design of consistent,mimetic schemes for these involutionary PDEs. The importance ofmultidimensional Riemann solvers in facilitating the design of such schemes isalso documented. We study the problem of curl constraint-preservingreconstruction as it pertains to the design of mimetic discontinuous Galerkin(DG) and finite volume (FV) schemes for PDEs that support such an involution.This is done for two and three dimensional structured mesh problems where wedeliver closed form expressions for the reconstruction. The role that thisreconstruction plays in the curl-free, or curl-preserving prolongation ofvector fields in adaptive mesh refinement (AMR) is also discussed. In twodimensions, a von Neumann analysis of structure-preserving DG-like schemes thatmimetically satisfy the curl constraints, is also presented. Numerical resultsare also presented to show that the schemes meet their design accuracy.

Curl constraint-preserving reconstruction and the guidance it gives for mimetic scheme design