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Error estimates to smooth solutions of semi‐discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws
Author(s) -
Huang Juntao,
Shu ChiWang
Publication year - 2017
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22089
Subject(s) - mathematics , monotone polygon , piecewise , gauss–laguerre quadrature , conservation law , quadrature (astronomy) , scalar (mathematics) , numerical integration , mathematical analysis , clenshaw–curtis quadrature , gaussian quadrature , tanh sinh quadrature , gauss–jacobi quadrature , galerkin method , gauss–kronrod quadrature formula , numerical analysis , nyström method , nonlinear system , geometry , boundary value problem , physics , quantum mechanics , electrical engineering , engineering
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, withP k(piecewise polynomials of degree k ) finite elements in 1D problems, if the quadrature over elements is exact for polynomials of degree ( 2 k ) , error estimates of O ( h k + 1 / 2 ) are obtained for general monotone fluxes, and optimal estimates of O ( h k + 1 ) are obtained for upwind fluxes. For multidimensional problems, if in addition quadrature over edges is exact for polynomials of degree ( 2 k + 1 ) , error estimates of O ( h k ) are obtained for general monotone fluxes, and O ( h k + 1 / 2 ) 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