ARBITRARY LAGRANGIAN-EULERIAN DISCONTINUOUS GALERKIN METHOD FOR HYPERBOLIC EQUATIONS INVOLVING delta-SINGULARITIES

摘要

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 kth 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 kth order, the convergence rate in H-(k+2) norm for the scheme with the upwind flux in the whole domain can achieve (k+ 1/2)th order, and the convergence rate in H-(k+1) (RR-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.