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

Meng, Xiong; Shu, Chi-Wang; Wu, Boying

doi:10.1090/mcom/3022

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

Math. Comp. 85 (2016), 1225-1261 Request permission

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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