Negative-Order Norm Estimates for Nonlinear Hyperbolic Conservation Laws

Authors:
Liangyue Ji;Yan Xu;Jennifer K. Ryan
Affiliations:
Department of Mathematics, University of Science and Technology of China, Hefei, People's Republic of China 230026 and School of Mathematics, University of Minnesota, Minneapolis, USA 55455;Department of Mathematics, University of Science and Technology of China, Hefei, People's Republic of China 230026;Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands 2628 CD
Venue:
Journal of Scientific Computing
Year:
2013

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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.