Discontinuous Galerkin method for hyperbolic equations involving \(\delta \) singularities: negativeorder norm error estimates and applications
 Yang Yang,
 ChiWang Shu
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In this paper, we develop and analyze discontinuous Galerkin (DG) methods to solve hyperbolic equations involving \(\delta \) singularities. Negativeorder norm error estimates for the accuracy of DG approximations to \(\delta \) singularities are investigated. We first consider linear hyperbolic conservation laws in one space dimension with singular initial data. We prove that, by using piecewise \(k\) th degree polynomials, at time \(t\) , the error in the \(H^{(k+2)}\) norm over the whole domain is \((k+1/2)\) th order, and the error in the \(H^{(k+1)}(\mathbb R \backslash \mathcal R _t)\) norm is \((2k+1)\) th order, where \(\mathcal R _t\) is the pollution region due to the initial singularity with the width of order \(\mathcal O (h^{1/2} \log (1/h))\) and \(h\) is the maximum cell length. As an application of the negativeorder norm error estimates, we convolve the numerical solution with a suitable kernel which is a linear combination of Bsplines, to obtain \(L^2\) error estimate of \((2k+1)\) th order for the postprocessed solution. Moreover, we also obtain high order superconvergence error estimates for linear hyperbolic conservation laws with singular source terms by applying Duhamel’s principle. Numerical examples including an acoustic equation and the nonlinear rendezvous algorithms are given to demonstrate the good performance of DG methods for solving hyperbolic equations involving \(\delta \) singularities.
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 Title
 Discontinuous Galerkin method for hyperbolic equations involving \(\delta \) singularities: negativeorder norm error estimates and applications
 Journal

Numerische Mathematik
Volume 124, Issue 4 , pp 753781
 Cover Date
 20130801
 DOI
 10.1007/s0021101305268
 Print ISSN
 0029599X
 Online ISSN
 09453245
 Publisher
 Springer Berlin Heidelberg
 Additional Links
 Topics
 Keywords

 65M60
 65M15
 Industry Sectors
 Authors

 Yang Yang ^{(1)}
 ChiWang Shu ^{(1)}
 Author Affiliations

 1. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA