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Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations


Authors: Xiong Meng, Chi-Wang Shu and Boying Wu
Journal: Math. Comp. 85 (2016), 1225-1261
MSC (2010): Primary 65M60, 65M12, 65M15
DOI: https://doi.org/10.1090/mcom/3022
Published electronically: September 2, 2015
MathSciNet review: 3454363
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Abstract: We analyze discontinuous Galerkin methods using upwind-biased numerical fluxes for time-dependent linear conservation laws. In one dimension, optimal a priori error estimates of order $ k+1$ are obtained for the semidiscrete scheme when piecewise polynomials of degree at most $ k$ $ (k \ge 0)$ are used. Our analysis is valid for arbitrary nonuniform regular meshes and for both periodic boundary conditions and for initial-boundary value problems. We extend the analysis to the multidimensional case on Cartesian meshes when piecewise tensor product polynomials are used, and to the fully discrete scheme with explicit Runge-Kutta time discretization. Numerical experiments are shown to demonstrate the theoretical results.


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Additional Information

Xiong Meng
Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China
Email: xiongmeng@hit.edu.cn

Chi-Wang Shu
Affiliation: Division of Applied Mathematics, Brown University, Providence, RI 02912
Email: shu@dam.brown.edu

Boying Wu
Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China
Email: mathwby@hit.edu.cn

DOI: https://doi.org/10.1090/mcom/3022
Keywords: Discontinuous Galerkin method, upwind-biased flux, conservation laws, error estimates
Received by editor(s): August 22, 2013
Received by editor(s) in revised form: October 30, 2014, and November 23, 2014
Published electronically: September 2, 2015
Additional Notes: The research of the second author was supported by NSF grants DMS-1112700 and DMS-1418750, and by DOE grant DE-FG02-08ER25863
Article copyright: © Copyright 2015 American Mathematical Society