Negative-order norm estimates for nonlinear hyperbolic conservation laws
Negative-order norm estimates for the accuracy of discontinuous Galerkin (DG) approximations to scalar nonlinear hyperbolic equations with smooth solutions are established. For these special solutions, this ``hidden accuracy'' can be extracted 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, now it is demonstrated that it is possible to extend the smoothness-increasing accuracy-conserving filter for scalar nonlinear hyperbolic equations. Furthermore, theoretical error estimates are provided for the DG solutions that show an improvement to \((2k+m)\)-th order in the negative-order norm, where \(m\) depends upon the chosen flux.
smoothness-increasing accuracy-conserving filter
scientific article; zbMATH DE number 6151379