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Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: Theory and boundary conditions

Authors:

Travis C. Fisher,

Mark H. Carpenter,

Jan NordströM,

Nail K. Yamaleev,

Charles SwansonAuthors Info & Claims

Journal of Computational Physics, Volume 234

Pages 353 - 375

https://doi.org/10.1016/j.jcp.2012.09.026

Published: 01 February 2013 Publication History

Abstract

The Lax-Wendroff theorem stipulates that a discretely conservative operator is necessary to accurately capture discontinuities. The discrete operator, however, need not be derived from the divergence form of the continuous equations. Indeed, conservation law equations that are split into linear combinations of the divergence and product rule form and then discretized using any diagonal-norm skew-symmetric summation-by-parts (SBP) spatial operator, yield discrete operators that are conservative. Furthermore, split-form, discretely conservation operators can be derived for periodic or finite-domain SBP spatial operators of any order. Examples are presented of a fourth-order, SBP finite-difference operator with second-order boundary closures. Sixth- and eighth-order constructions are derived, and are supplied in an accompanying text file.

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Published In

cover image Journal of Computational Physics

Journal of Computational Physics Volume 234, Issue

February, 2013

574 pages

ISSN:0021-9991

Issue’s Table of Contents

Copyright © © 2012.

Publisher

Academic Press Professional, Inc.

United States

Publication History

Published: 01 February 2013

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Cited By

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https://dl.acm.org/doi/10.1016/j.jcp.2024.113360
Cicchino ANadarajah S(2024)Scalable evaluation of Hadamard products with tensor product basis for entropy-stable high-order methodsJournal of Computational Physics10.1016/j.jcp.2024.113134513:COnline publication date: 15-Sep-2024
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Glaubitz JKlein SNordström JÖffner P(2024)Summation-by-parts operators for general function spacesJournal of Computational Physics10.1016/j.jcp.2024.112889504:COnline publication date: 1-May-2024
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Edoh A(2024)Conservative correction procedures utilizing artificial dissipation operatorsJournal of Computational Physics10.1016/j.jcp.2024.112880504:COnline publication date: 1-May-2024
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Mantri YÖffner PRicchiuto M(2024)Fully well-balanced entropy controlled discontinuous Galerkin spectral element method for shallow water flowsJournal of Computational Physics10.1016/j.jcp.2023.112673498:COnline publication date: 1-Feb-2024
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