Superconvergence of Discontinuous Galerkin Methods for Linear Hyperbolic Equations with Singular Initial Data
Year: 2017
International Journal of Numerical Analysis and Modeling, Vol. 14 (2017), Iss. 3 : pp. 342–354
Abstract
In this paper, we consider the discontinuous Galerkin (DG) methods to solve linear hyperbolic equations with singular initial data. With the help of weight functions, the superconvergence properties outside the pollution region will be investigated. We show that, by using piecewise polynomials of degree $k$ and suitable initial discretizations, the DG solution is $(2k+1)$-th order accurate at the downwind points and $(k+2)$-th order accurate at all the other downwind-biased Radau points. Moreover, the derivative of error between the DG and exact solutions converges at a rate of $k+1$ at all the interior upwind-biased Radau points. Besides the above, the DG solution is also $(k+2)$-th order accurate towards a particular projection of the exact solution and the numerical cell averages are $(2k+1)$-th order accurate. Numerical experiments are presented to confirm the theoretical results.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2017-IJNAM-10011
International Journal of Numerical Analysis and Modeling, Vol. 14 (2017), Iss. 3 : pp. 342–354
Published online: 2017-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 13
Keywords: Discontinuous Galerkin (DG) method singular initial data linear hyperbolic equations superconvergence weight function weighted norms.