This paper examines a class of involutionconstrained 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 curlfree or curlpreserving, respectively. They arise very frequently in equations for hyperelasticity and compressible multiphase flow, in certain formulations of general relativity and in the numerical solution of Schrödinger’s equation. Experience has shown that if nothing special is done to account for the curlpreserving vector field, it can blow up in a finite amount of simulation time. In this paper, we catalogue a class of DGlike schemes for such PDEs. To retain the globally curlfree or curlpreserving constraints, the components of the vector field, as well as their higher moments, must be collocated at the edges of the mesh. They are updated using potentials collocated at the vertices of the mesh. The resulting schemes: (i) do not blow up even after very long integration times, (ii) do not need any special cleaning treatment, (iii) can operate with large explicit timesteps, (iv) do not require the solution of an elliptic system and (v) can be extended to higher orders using DGlike methods. Themore »
Several important PDE systems, like magnetohydrodynamics and computational electrodynamics, are known to support involutions where the divergence of a vector field evolves in divergencefree or divergence constraintpreserving fashion. Recently, new classes of PDE systems have emerged for hyperelasticity, compressible multiphase flows, socalled firstorder reductions of the Einstein field equations, or a novel firstorder hyperbolic reformulation of Schrödinger’s equation, to name a few, where the involution in the PDE supports curlfree or curl constraintpreserving evolution of a vector field. We study the problem of curl constraintpreserving reconstruction as it pertains to the design of mimetic finite volume (FV) WENOlike schemes for PDEs that support a curlpreserving involution. (Some insights into discontinuous Galerkin (DG) schemes are also drawn, though that is not the prime focus of this paper.) This is done for two and threedimensional structured mesh problems where we deliver closed form expressions for the reconstruction. The importance of multidimensional Riemann solvers in facilitating the design of such schemes is also documented. In two dimensions, a von Neumann analysis of structurepreserving WENOlike schemes that mimetically satisfy the curl constraints, is also presented. It shows the tremendous value of higher order WENOlike schemes in minimizing dissipation and dispersion for this class more »
 Publication Date:
 NSFPAR ID:
 10360536
 Journal Name:
 Communications on Applied Mathematics and Computation
 Volume:
 5
 Issue:
 1
 Page Range or eLocationID:
 p. 235294
 ISSN:
 20966385
 Publisher:
 Springer Science + Business Media
 Sponsoring Org:
 National Science Foundation
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