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Accurate High-Order Tensor-Product Generalized Summation-By-Parts Discretizations of Hyperbolic Conservation Laws: General Curved Domains and Functional Superconvergence

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Abstract

The goal of this paper is to outline the requirements for obtaining accurate solutions and functionals from high-order tensor-product generalized summation-by-parts discretizations of the steady two-dimensional linear convection and Euler equations on general curved domains. Two procedures for constructing high-order grids using either Lagrange or B-spline mappings are outlined. For the linear convection equation, four discretizations are derived and characterized—two based on the mortar-element approach and two based on the global summation-by-parts-operator approach. It is shown numerically that the schemes are dual consistent, and the requirements for achieving functional superconvergence for each set of methods are outlined. For the Euler equations, a dual-consistent mortar-element discretization is proposed and the practical requirements for obtaining accurate solutions and superconvergent functionals for problems of increasing practical relevance are delineated through theory and numerical examples.

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Acknowledgements

A portion of the plots appearing in this paper were created using Matplotlib [19].

Funding

This work was supported by the Natural Sciences and Engineering Research Council of Canada, the Government of Ontario, and the University of Toronto. A portion of the computations were performed on the Niagara supercomputer at the SciNet HPC Consortium [21]. SciNet is funded by: the Canada Foundation for Innovation; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto.

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  1. University of Toronto Institute for Aerospace Studies, Toronto, Canada

    David A. Craig Penner & David W. Zingg

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  1. David A. Craig Penner
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  2. David W. Zingg
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Contributions

David A. Craig Penner: Conceptualization, Methodology, Software, Writing – original draft; David W. Zingg: Conceptualization, Methodology, Supervision, Writing – review & editing.

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Correspondence to David A. Craig Penner.

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Craig Penner, D.A., Zingg, D.W. Accurate High-Order Tensor-Product Generalized Summation-By-Parts Discretizations of Hyperbolic Conservation Laws: General Curved Domains and Functional Superconvergence. J Sci Comput 93, 36 (2022). https://doi.org/10.1007/s10915-022-01990-w

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