Arbitrary Lagrangian--Eulerian Discontinuous Galerkin Method for Hyperbolic Equations Involving $\delta$-Singularities

Abstract

In this paper, we develop and analyze an arbitrary Lagrangian--Eulerian discontinuous Galerkin (ALE-DG) method for solving one-dimensional hyperbolic equations involving $\delta$-singularities on moving meshes. The $L^2$ and negative norm error estimates are proven for the ALE-DG approximation. More precisely, when choosing the approximation space with piecewise $k$th degree polynomials, the convergence rate in $L^2$-norm for the scheme with the upwind numerical flux is $(k+1)$th order in the region apart from the singularities, the convergence rate in $H^{-(k+1)}$ norm for the scheme with the monotone fluxes in the whole domain is $k$th order, the convergence rate in $H^{-(k+2)}$ norm for the scheme with the upwind flux in the whole domain can achieve $(k+\frac{1}{2})$th order, and the convergence rate in $H^{-(k+1)}(R\backslash R_T)$ norm for the scheme with the upwind flux is $(2k+1)$th order, where $R_T$ is the pollution region at time $T$ due to the singularities. Moreover, numerically the $(2k+1)$th order accuracy for the postprocessed solution in the smooth region can be obtained, which is produced by convolving the ALE-DG solution with a suitable kernel consisting of B-splines. Numerical examples are shown to demonstrate the accuracy and capability of the ALE-DG method for the hyperbolic equations involving $\delta$-singularity on moving meshes.