Discontinuous Galerkin method for hyperbolic equations involving $$\delta $$ -singularities: negative-order norm error estimates and applications

Abstract

In this paper, we develop and analyze discontinuous Galerkin (DG) methods to solve hyperbolic equations involving $$\delta $$ -singularities. Negative-order 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 negative-order norm error estimates, we convolve the numerical solution with a suitable kernel which is a linear combination of B-splines, to obtain $$L^2$$ error estimate of $$(2k+1)$$ th order for the post-processed 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 rendez-vous algorithms are given to demonstrate the good performance of DG methods for solving hyperbolic equations involving $$\delta $$ -singularities.