Communications on Applied Mathematics and Computation >

2023 , Vol. 5 >Issue 1: 235 - 294

DOI: https://doi.org/10.1007/s42967-021-00160-3

ORIGINAL PAPERS

Curl Constraint-Preserving Reconstruction and the Guidance it Gives for Mimetic Scheme Design

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  • 1 Physics and ACMS Departments, University of Notre Dame, Notre Dame, IN, USA;
    2 Seminar for Applied Mathematics(SAM), Department of Mathematics, ETH Zürich, CH-8092 Zurich, Switzerland;
    3 Department of Mathematics and Computer Science, University of Ferrara, Ferrara, Italy;
    4 Laboratory of Applied Mathematics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italy

Received date: 2021-03-21

  Revised date: 2021-06-09

  Online published: 2023-03-08

Supported by

Dinshaw S. Balsara acknowledges support via NSF grants NSF-19-04774, NSFAST-2009776 and NASA-2020-1241. Michael Dumbser acknowledges the financial support received from the Italian Ministry of Education, University and Research (MIUR) in the frame of the Departments of Excellence Initiative 2018–2022 attributed to DICAM of the University of Trento (grant L. 232/2016) and in the frame of the PRIN 2017 project Innovative numerical methods for evolutionary partial differential equations and applications. Michael Dumbser has also received funding from the University of Trento via the Strategic Initiative Modeling and Simulation. Walter Boscheri acknowledges funding from the Istituto Nazionale di Alta Matematica (INdAM) through the GNCS group and the program Young Researchers Funding 2018 via the research project semi-implicit structure-preserving schemes for continuum mechanics.

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Abstract

Several important PDE systems, like magnetohydrodynamics and computational electrodynamics, are known to support involutions where the divergence of a vector field evolves in divergence-free or divergence constraint-preserving fashion. Recently, new classes of PDE systems have emerged for hyperelasticity, compressible multiphase flows, so-called firstorder reductions of the Einstein field equations, or a novel first-order hyperbolic reformulation of Schrödinger’s equation, to name a few, where the involution in the PDE supports curl-free or curl constraint-preserving evolution of a vector field. We study the problem of curl constraint-preserving reconstruction as it pertains to the design of mimetic finite volume (FV) WENO-like schemes for PDEs that support a curl-preserving 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 three-dimensional 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 structure-preserving WENOlike schemes that mimetically satisfy the curl constraints, is also presented. It shows the tremendous value of higher order WENO-like schemes in minimizing dissipation and dispersion for this class of problems. Numerical results are also presented to show that the edge-centered curl-preserving (ECCP) schemes meet their design accuracy. This paper is the first paper that invents non-linearly hybridized curl-preserving reconstruction and integrates it with higher order Godunov philosophy. By its very design, this paper is, therefore, intended to be forward-looking and to set the stage for future work on curl involution-constrained PDEs.

Key words： PDEs; Numerical schemes; Mimetic

Cite this article

Dinshaw S. Balsara, Roger Käppeli, Walter Boscheri, Michael Dumbser . Curl Constraint-Preserving Reconstruction and the Guidance it Gives for Mimetic Scheme Design[J]. Communications on Applied Mathematics and Computation, 2023 , 5(1) : 235 -294 . DOI: 10.1007/s42967-021-00160-3

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