Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations
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- by Xiong Meng, Chi-Wang Shu and Boying Wu PDF
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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.References
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Additional Information
- Xiong Meng
- Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China
- MR Author ID: 998988
- Email: xiongmeng@hit.edu.cn
- Chi-Wang Shu
- Affiliation: Division of Applied Mathematics, Brown University, Providence, RI 02912
- MR Author ID: 242268
- Email: shu@dam.brown.edu
- Boying Wu
- Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China
- MR Author ID: 261930
- Email: mathwby@hit.edu.cn
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 85 (2016), 1225-1261
- MSC (2010): Primary 65M60, 65M12, 65M15
- DOI: https://doi.org/10.1090/mcom/3022
- MathSciNet review: 3454363