Abstract
In this paper, we establish negative-order norm estimates for the accuracy of discontinuous Galerkin (DG) approximations to scalar nonlinear hyperbolic equations with smooth solutions. For these special solutions, we are able to extract this “hidden accuracy” through the use of a convolution kernel that is composed of a linear combination of B-splines. Previous investigations into extracting the superconvergence of DG methods using a convolution kernel have focused on linear hyperbolic equations. However, we now demonstrate that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter for scalar nonlinear hyperbolic equations. Furthermore, we provide theoretical error estimates for the DG solutions that show improvement to \((2k+m)\)-th order in the negative-order norm, where \(m\) depends upon the chosen flux.
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Acknowledgments
The second authors research is supported by NSFC Grant No.10971211, No.11031007, FANEDD No.200916, NCET No.09-0922, Fok Ying Tung Education Foundation No.131003. Additional support is provided by the Alexander von Humboldt-Foundation while the author was in residence at Freiburg University, Germany. The third authors research is supported by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under Grant number FA8655-09-1-3055
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Dedicated to Professor Stanley Osher on the occasion of his 70th birthday.
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Ji, L., Xu, Y. & Ryan, J.K. Negative-Order Norm Estimates for Nonlinear Hyperbolic Conservation Laws. J Sci Comput 54, 531–548 (2013). https://doi.org/10.1007/s10915-012-9668-6
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DOI: https://doi.org/10.1007/s10915-012-9668-6