TY - JOUR

T1 - A posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional nonlinear scalar conservation laws

AU - Baccouch, Mahboub

N1 - Funding Information:
This research was supported by the NASA Nebraska Space Grant Program (Grant NNX10AN62H ) and the University Committee on Research and Creative Activity (UCRCA Proposal 2012-06-F ) at the University of Nebraska at Omaha.

PY - 2014/10

Y1 - 2014/10

N2 - In this paper, new a posteriori error estimates for a discontinuous Galerkin (DG) formulation applied to nonlinear scalar conservation laws in one space dimension are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local problem with no boundary condition on each element of the mesh. We first show that the leading error term on each element for the solution is proportional to a (p+1)-degree Radau polynomial, when p-degree piecewise polynomials with p≤1 are used. This result allows us to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L2-norm under mesh refinement. The order of convergence is proved to be p+5/4. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h1/2) rate. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates.

AB - In this paper, new a posteriori error estimates for a discontinuous Galerkin (DG) formulation applied to nonlinear scalar conservation laws in one space dimension are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local problem with no boundary condition on each element of the mesh. We first show that the leading error term on each element for the solution is proportional to a (p+1)-degree Radau polynomial, when p-degree piecewise polynomials with p≤1 are used. This result allows us to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L2-norm under mesh refinement. The order of convergence is proved to be p+5/4. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at O(h1/2) rate. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates.

KW - A posteriori error estimation

KW - Discontinuous Galerkin method

KW - Nonlinear conservation laws

KW - Superconvergence

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U2 - 10.1016/j.apnum.2014.04.001

DO - 10.1016/j.apnum.2014.04.001

M3 - Article

AN - SCOPUS:84901940202

SN - 0168-9274

VL - 84

SP - 1

EP - 21

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

ER -