Error estimates to smooth solutions of semi-discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws
Abstract
In this article, we focus on error estimates to smooth solutions of semi‐discrete discontinuous Galerkin (DG) methods with quadrature rules for scalar conservation laws. The main techniques we use are energy estimate and Taylor expansion first introduced by Zhang and Shu in (Zhang and Shu, SIAM J Num Anal 42 (2004), 641–666). We show that, with (piecewise polynomials of degree k ) finite elements in 1D problems, if the quadrature over elements is exact for polynomials of degree , error estimates of are obtained for general monotone fluxes, and optimal estimates of are obtained for upwind fluxes. For multidimensional problems, if in addition quadrature over edges is exact for polynomials of degree , error estimates of are obtained for general monotone fluxes, and are obtained for monotone and sufficiently smooth numerical fluxes. Numerical results validate our analysis. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 467–488, 2017
- Authors:
-
- Tsinghua Univ., Beijing (China)
- Brown Univ., Providence, RI (United States)
- Publication Date:
- Research Org.:
- Brown Univ., Providence, RI (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC); National Science Foundation (NSF)
- OSTI Identifier:
- 1533199
- Alternate Identifier(s):
- OSTI ID: 1401007
- Grant/Contract Number:
- FG02-08ER25863; DMS-1418750; DE‐FG02‐08ER25863
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Numerical Methods for Partial Differential Equations
- Additional Journal Information:
- Journal Volume: 33; Journal Issue: 2; Journal ID: ISSN 0749-159X
- Publisher:
- Wiley
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; mathematics; Discontinuous Galerkin; error estimate; quadrature rules; conservation laws; semi-discrete
Citation Formats
Huang, Juntao, and Shu, Chi‐Wang. Error estimates to smooth solutions of semi-discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws. United States: N. p., 2016.
Web. doi:10.1002/num.22089.
Huang, Juntao, & Shu, Chi‐Wang. Error estimates to smooth solutions of semi-discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws. United States. https://doi.org/10.1002/num.22089
Huang, Juntao, and Shu, Chi‐Wang. Tue .
"Error estimates to smooth solutions of semi-discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws". United States. https://doi.org/10.1002/num.22089. https://www.osti.gov/servlets/purl/1533199.
@article{osti_1533199,
title = {Error estimates to smooth solutions of semi-discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws},
author = {Huang, Juntao and Shu, Chi‐Wang},
abstractNote = {In this article, we focus on error estimates to smooth solutions of semi‐discrete discontinuous Galerkin (DG) methods with quadrature rules for scalar conservation laws. The main techniques we use are energy estimate and Taylor expansion first introduced by Zhang and Shu in (Zhang and Shu, SIAM J Num Anal 42 (2004), 641–666). We show that, with (piecewise polynomials of degree k ) finite elements in 1D problems, if the quadrature over elements is exact for polynomials of degree , error estimates of are obtained for general monotone fluxes, and optimal estimates of are obtained for upwind fluxes. For multidimensional problems, if in addition quadrature over edges is exact for polynomials of degree , error estimates of are obtained for general monotone fluxes, and are obtained for monotone and sufficiently smooth numerical fluxes. Numerical results validate our analysis. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 467–488, 2017},
doi = {10.1002/num.22089},
journal = {Numerical Methods for Partial Differential Equations},
number = 2,
volume = 33,
place = {United States},
year = {Tue Aug 30 00:00:00 EDT 2016},
month = {Tue Aug 30 00:00:00 EDT 2016}
}
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